elementary statisticselementary statistics chapter 06 dr. ghamsary page 5 5 sample size the...

Elementary Statistics Chapter 06 Dr. Ghamsary

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Elementary Statistics

M. Ghamsary, Ph.D.

Chapter 06

Confidence Interval


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Confidence Interval

• Point Estimate

• Interval Estimate Point Estimate: a single specific value, usually from data, used to approximate a population

parameter.

1. Mean: x is the best point estimate of µ

2. Proportion: p is the best point estimate of p

3. Variance: s 2 is the best point estimate of σ 2

4. Standard deviation: s is a point estimate of σ (Note that this is a biased estimator)

Interval Estimate (Confidence Interval):

A confidence interval gives an estimated range of values which is likely to include an unknown

population parameter, the estimated range being calculated from a given set of sample data.

If independent samples are taken repeatedly from the same population, and a confidence interval

calculated for each sample, then a certain percentage (confidence level) of the intervals will include

the unknown population parameter. This interval has certain degree of confidence, usually it is called

confidence level. The most common choices for confidence level are 90%, 95%, and 99%.

The width of the confidence interval gives us some idea about how uncertain we are about the

unknown parameter (the narrower it is the more precision it has). A very wide interval may indicate

that more data should be collected before anything very definite can be said about the parameter.

Confidence intervals are more informative than the simple results of hypothesis tests (where we decide

"reject H0" or "don't reject H0". This will be discussed in the next chapter) since they provide a range

of plausible values for the unknown parameter

We denote the degree of confidence by (1-α )%, where α is called level of significant.


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Area in Tails and 2

zα Notation

0 05.z =1.645 (the Z-score which has 0.05 to the right, or 0.95 blow Z).

0 10.z = 1.282 (the Z-score which has 0.10 to the right, or 0.90 below Z).

The Greek letter alpha α is used represent the area in both tails for a confidence interval, and so

alpha/2 will be the area in one tail. These values are from Table I, at the end of this section.

Here are some common values

Confidence Level

Level of Significance

α

Are below z Area in One tail

2/α 2

zα

80% 0.20 0.9000 0.1000 1.282

90% 0.10 0.9500 0.0500 1.645

95% 0.05 0.9750 0.0250 1.960

98% 0.02 0.9900 0.0100 2.326

99% 0.01 0.9950 0.0050 2.576

Notice in the above table, that the area between 0 and the z-score is simply one-half of the

confidence level. So, if there is a confidence level which isn't given above, all you need to do to is

to find α divide by two, 2/α and then look up the area in the inside part of the and look up the z-

score on the outside.

Note: We use the assumption that the data collected is simple random sample (SRS)

The format of confidence interval is as follows: a < Population Parameter < b

For example:

• 12 25< <µ is a 95% confidence interval for population mean,

• 0 45 0 52. .< <p , is a 95% confidence interval for population proportion, and

• 1 65 12 64. .< <σ is a 95% confidence interval for population standard deviation.


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Confidence Interval for the Mean: Generally the data must come from a normal

distribution, particularly for small samples (n<30). However, in the case of larger than 30 we

may relax this assumption, by CLT.

A. If σ , the standard deviation of the population, is known, then we use

2

xn

Zασ

± , or x E± with 2

En

Zασ

= or

2 2

x xn n

Z Zα ασ σµ− < < + ,

Where2

En

Zασ

= and it is called margin of error.

B. If σ is unknown, but n >30 we can still use the above formula, even if we do not have a

normally distributed population (because we can apply CLT). Remember to replace σ bys

that is 2

xn

Z sα± . But I do not recommend this.

C. Ifσ is unknown, then we use 2

xn

t sα± , or x E± with

2E

nt sα= or

2 2

s sx xn

tn

tα αµ− < < +

Where 2

tα , is read from Student t distribution in Table II.

D. If we have small sample and the population is not normal, we cannot apply any of the

above Z or t formula. In a case like this, we recommend a nonparametric statistics.


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Sample Size The following formula is used to find the sample size to estimate the mean of the population.

2

2Z

nEα σ⎛ ⎞

⎜ ⎟=⎜ ⎟⎝ ⎠

,

Confidence Interval for the proportion: if we repeat an experiment n times

with x successes, then we have: ˆ xpn

= , which is called the sample proportion that estimate

the population proportion p .

The Confidence Interval is given by

p̂ E± , with 2

ˆZ q

nE

ˆpα= or

2 2

ˆ ˆˆ ˆ ˆ ˆp pZ Zp pn

pq qnα α− < < +

Remember the condition 5np ≥ and 5nq ≥ or 10npq ≥

Sample Size:

2

2Z

ˆ ˆn pqEα⎛ ⎞

⎜ ⎟=⎜ ⎟⎝ ⎠

, where q p= −1

Remark: If you have no idea or no prior knowledge of p , then use 0 50ˆ ˆ .p q= = .

Student t distribution

It is often the case that one wants to calculate the size of sample needed to obtain a certain level of

confidence in survey results. Unfortunately, this calculation requires prior knowledge of the

population standard deviationσ . Realistically,σ is unknown. Often a preliminary sample will be

conducted so that a reasonable estimate of this critical population parameter can be made. If such a

preliminary sample is not made, but confidence intervals for the population mean are to be


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constructing using an unknownσ , then the distribution known as the Student t distribution can be

used.

First let us have a little history about this curious name. William Gosset (1876-1937) was a Guinness

Brewery employee who needed a distribution that could be used with small samples. Since the Irish

brewery did not allow publication of research results, he published under the pseudonym of Student.

We know that large samples approach a normal distribution. What Gosset showed was that small

samples taken from an essentially normal population have a distribution characterized by the sample

size. The population does not have to be exactly normal, only unimodal and basically symmetric. This

is often characterized as heap-shaped or mound shaped.

Following are the important properties of the Student t distribution.

1. The Student t distribution is different for different sample sizes.

2. The Student t distribution is generally bell-shaped, but with smaller sample sizes shows

increased variability (flatter). In other words, the distribution is less peaked than a

normal distribution and with thicker tails. As the sample size increases, the distribution

approaches a normal distribution. For n > 30, the differences are negligible.

3. The mean is zero (much like the standard normal distribution).

4. The distribution is symmetrical about the mean.

5. The variance is greater than one, but approaches one from above as the sample size

increases (σ =1 for the standard normal distribution).

6. The population standard deviation is unknown.

7. The population is essentially normal (unimodal and basically symmetric)


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Confidence Interval for Variance Standard Deviation: Again the data

must come from a normal distribution for small samples.

( ) ( )2 22

22

1 1

LR

n s n sσ

χχ− −

< < ,

Where, 2 21 2

L αχ χ−

= is the cutoff for the Right tail of chi-square, 2 2

2R αχ χ= is the cutoff for the Left

tail of chi-square. Where

o n: Number of observations in the data set

o s is the standard deviation of the sample

o σ : is the standard deviation of the population

Chi-Square Distribution

The population variance (and its square root, the population standard deviation) is a measure of the

variability (or, if you like, the uniformity) of the population. It is not as common to require an estimate

of σ or σ2 in statistical work as it is to require an estimate of the population mean or population

proportion. In fact, if anything, the comparison of two population variances is a more common

requirement than the estimation of the variance of a single population (and we will deal with that

comparison of variances later as part of our study of hypothesis testing). Still, of the population

parameters that require a sampling distribution which is non-symmetric, the variance is the most

important. So, we will look briefly at the estimation of the population variance so you know how to

compute interval estimates of the population variance and population standard deviation, and so you

know how to work with interval estimates when the sampling distribution involved is not symmetric.

We know from the brief comment in the document on sampling distributions that when the population

is approximately normally distributed, the variable

( ) 22

2

1n sχ

σ−

=


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has the so-called chi-square, 2χ -distribution, with df= n - 1 degrees of freedom

Example 1A: Find the values of2

Zα , 2

tα , 2

2αχ and 2

12

αχ − for the α = 0 10. and n =21.

Solution:

2

Zα : 2

1 645Z .α = ± we do not need n, we go directly to Z table an look under 0.1/2=0.05.

2

tα : we need n and df=n-1=21-1=20. Then from Table II under α = 0 10. two tails we will get

21 725t .α = ±

2

2αχ : 2 2

0 1 0 052

31 410. . .χ χ == we need n and Table III,

21 2αχ−

: 2 2 20 1 1 0 05 0 951 2

10 851. . . .χ χ χ−−= ==

Example 1B: Find the values of2

Zα , 2

tα , 2

2αχ and 2

12

αχ − for the 0 01.α = and n =16.


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Zα , 2

tα , 2

2αχ and 2

12

αχ − for the 95% confidence level and n =15.

Solution: We will use α = =5% 0 05. , since we need 95% confidence interval.

2

Zα : Again we go to Table I and we get 2

1 96Z .α = ± , by looking at 0.05/2=0.025.

2

tα : Just like Example 1 we go to Table II with df=15-1=14 to get 2

2 145t .α = ±

2

2αχ : 2 2

0 05 0 0252


21 2αχ−

: 2 2 20 05 1 0 025 0 9751 2

5 629. . . .χ χ χ−−= = =


Zα , 2

tα , 2

2αχ and 2

12



10


Zα , 2

tα , 2

2αχ and 2

12


Solution: We must use α = =2% 0 02. , since we need 98% confidence interval.

2

Zα :We will get 2

2 33Z .α = ± , since 0 012 .α = and from the Table I the which gives Z=2.33.

2

tα : Just like before we go to Table II with df=35-1=34 to get 2

2 441t .α = ±

Since n>30 we use either t or z table. So 2 2

2 326t Z .α α≈ =

2

2αχ : 2 2

0 02 0 012


21 2αχ−

: 2 2 20 02 1 0 01 0 991 2

17 789. . . .χ χ χ−−= ==


Zα , 2

tα , 2

2αχ and 2

12



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Example 4A: A random sample of 25 students at UCLA yields 2 80.x = and assume 0 60.σ = .

Construct the 95% confidence interval for the true population mean,µ . Also assume data is coming

from a normal population.

Solution: Since σ is given we use2

E sZnα= to find the margin of error.

2

0 60 0 14985 0 2352

1 95

6sZ . .E .n

.α= = = ≈

So we will have 2 80 0 235. .x E± = ± or 2.80-0.235 < µ <2.80+0.235

2.565 < µ <3.035.

Example 4B: A random sample of 25 students at UCLA yields 2 80.x = and assume 0 60.s = .

Construct the 95% confidence interval for the true population mean,µ . Also assume data is coming

from a normal population.

Example 5: A random sample of 64 students at UCG yields x = 2 94. and s = 0 60. . Construct the 95%

confidence interval for the true population mean,µ .

Solution: Since σ is not given we use2

En

t sα= to find the margin of

error.2

0 60 0 14985 0 1564

1 998t .s .E . .nα= = = ≈ and

x E± = ±2 94 0 15. . or 2.94-0.15 < µ <2.94+0.15 2.79 < µ <3.09.

Or we can use z formula (Since sample is larger than 30)

2

0 601 96 0 1564.. .Z sE

nα= = ≈ and x E± = ±2 94 0 15. . or 2.94-0.15 < µ <2.94+0.15

2.79 < µ <3.09


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Very Important: The above statement is not saying that 95% of the students have GPA between 2.79 and 3.09.

But the chance is 95%, not the probability, that the interval contains the µ. In other word, if we

construct 100 such intervals, approximately 95 of them should include the true value of the

population mean.

Example 6: In following we have randomly generated 100 confidence intervals (95% level) for the

mean of an exponential with the population mean of 5. Variable N Mean StDev SEM 95.0 % CI

C1 50 5.238 4.622 0.707 ( 3.852, 6.624) C2 50 4.439 4.894 0.707 ( 3.053, 5.825) C3 50 4.328 3.654 0.707 ( 2.942, 5.714) C4 50 4.852 5.218 0.707 ( 3.466, 6.238) C5 50 4.407 4.083 0.707 ( 3.021, 5.793) C6 50 5.125 4.971 0.707 ( 3.739, 6.511) C7 50 5.871 6.789 0.707 ( 4.485, 7.257) C8 50 4.024 4.603 0.707 ( 2.638, 5.410) C9 50 5.568 6.056 0.707 ( 4.181, 6.954) C10 50 4.778 4.398 0.707 ( 3.392, 6.164) C11 50 4.433 5.092 0.707 ( 3.047, 5.819) C12 50 4.873 3.933 0.707 ( 3.487, 6.259) C13 50 4.167 4.570 0.707 ( 2.781, 5.553) C14 50 5.471 4.254 0.707 ( 4.085, 6.857) C15 50 4.733 6.562 0.707 ( 3.347, 6.120) C16 50 4.892 4.501 0.707 ( 3.506, 6.278) C17 50 5.056 3.862 0.707 ( 3.670, 6.442) C18 50 5.440 5.135 0.707 ( 4.054, 6.826) C19 50 4.608 4.125 0.707 ( 3.222, 5.994) C20 50 5.978 8.269 0.707 ( 4.592, 7.364) C21 50 5.807 6.781 0.707 ( 4.421, 7.193) C22 50 5.197 5.031 0.707 ( 3.810, 6.583) C23 50 3.634 3.995 0.707 ( 2.248, 5.020) C24 50 5.301 5.076 0.707 ( 3.915, 6.687) C25 50 6.154 5.291 0.707 ( 4.768, 7.540) C26 50 4.978 5.032 0.707 ( 3.592, 6.364) C27 50 4.348 3.987 0.707 ( 2.961, 5.734) C28 50 6.854 8.811 0.707 ( 5.468, 8.240) C29 50 4.992 5.236 0.707 ( 3.605, 6.378) C30 50 4.904 5.419 0.707 ( 3.518, 6.290) C31 50 3.912 3.610 0.707 ( 2.526, 5.298) C32 50 4.788 4.714 0.707 ( 3.402, 6.174) C33 50 4.031 3.120 0.707 ( 2.645, 5.418) C34 50 4.335 3.914 0.707 ( 2.949, 5.721) C35 50 4.828 4.244 0.707 ( 3.442, 6.214) C36 50 5.529 5.462 0.707 ( 4.143, 6.915) C37 50 5.828 5.031 0.707 ( 4.442, 7.214) C38 50 5.038 4.911 0.707 ( 3.652, 6.424) C39 50 4.878 4.369 0.707 ( 3.492, 6.264) C40 50 5.970 6.844 0.707 ( 4.584, 7.357) C41 50 5.353 5.129 0.707 ( 3.967, 6.739) C42 50 4.826 4.255 0.707 ( 3.440, 6.212)


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C43 50 5.569 4.554 0.707 ( 4.183, 6.955) C44 50 6.540 7.035 0.707 ( 5.154, 7.926) C45 50 5.185 5.313 0.707 ( 3.799, 6.571) C46 50 4.406 3.623 0.707 ( 3.020, 5.792) C47 50 4.531 4.843 0.707 ( 3.145, 5.917) C48 50 4.581 4.153 0.707 ( 3.195, 5.967) C49 50 4.599 4.672 0.707 ( 3.212, 5.985) C50 50 5.810 5.864 0.707 ( 4.424, 7.196) C51 50 5.123 5.075 0.707 ( 3.737, 6.509) C52 50 5.228 4.306 0.707 ( 3.842, 6.614) C53 50 4.888 4.280 0.707 ( 3.502, 6.274) C54 50 6.139 6.433 0.707 ( 4.753, 7.525) C55 50 4.877 4.843 0.707 ( 3.491, 6.263) C56 50 5.175 4.501 0.707 ( 3.789, 6.562) C57 50 5.300 3.940 0.707 ( 3.914, 6.686) C58 50 3.935 4.002 0.707 ( 2.549, 5.321) C59 50 5.134 4.302 0.707 ( 3.748, 6.520) C60 50 4.899 5.267 0.707 ( 3.513, 6.285) C61 50 4.539 3.456 0.707 ( 3.153, 5.925) C62 50 4.305 4.092 0.707 ( 2.919, 5.691) C63 50 4.922 4.407 0.707 ( 3.536, 6.308) C64 50 5.008 5.798 0.707 ( 3.622, 6.394) C65 50 5.135 4.280 0.707 ( 3.749, 6.521) C66 50 4.589 4.448 0.707 ( 3.203, 5.975) C67 50 4.858 3.839 0.707 ( 3.472, 6.244) C68 50 4.607 5.223 0.707 ( 3.221, 5.993) C69 50 5.012 4.180 0.707 ( 3.625, 6.398) C70 50 5.513 7.519 0.707 ( 4.127, 6.899) C71 50 5.501 4.599 0.707 ( 4.114, 6.887) C72 50 4.772 5.200 0.707 ( 3.386, 6.158) C73 50 4.978 5.423 0.707 ( 3.592, 6.364) C74 50 4.758 3.664 0.707 ( 3.372, 6.144) C75 50 4.039 4.548 0.707 ( 2.653, 5.425) C76 50 3.946 3.718 0.707 ( 2.559, 5.332) C77 50 5.906 5.182 0.707 ( 4.520, 7.292) C78 50 5.704 5.451 0.707 ( 4.318, 7.090) C79 50 6.690 7.379 0.707 ( 5.304, 8.076) C80 50 4.364 4.111 0.707 ( 2.978, 5.750) C81 50 4.741 5.498 0.707 ( 3.355, 6.127) C82 50 6.809 6.483 0.707 ( 5.423, 8.195) C83 50 3.987 4.239 0.707 ( 2.601, 5.373) C84 50 5.783 5.160 0.707 ( 4.397, 7.169) C85 50 4.508 6.261 0.707 ( 3.122, 5.894) C86 50 4.854 6.486 0.707 ( 3.468, 6.240) C87 50 3.880 4.591 0.707 ( 2.494, 5.267) C88 50 5.880 6.288 0.707 ( 4.494, 7.266) C89 50 5.062 6.379 0.707 ( 3.676, 6.448) C90 50 4.718 5.273 0.707 ( 3.332, 6.104) C91 50 5.130 6.101 0.707 ( 3.744, 6.516) C92 50 4.715 3.879 0.707 ( 3.329, 6.101) C93 50 4.775 5.056 0.707 ( 3.389, 6.161) C94 50 5.723 4.971 0.707 ( 4.337, 7.109) C95 50 5.143 5.047 0.707 ( 3.757, 6.529) C96 50 4.661 3.996 0.707 ( 3.275, 6.048)

C97 50 5.135 4.771 0.707 ( 3.749, 6.521) C98 50 6.265 5.646 0.707 ( 4.878, 7.651)


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C99 50 4.707 3.629 0.707 ( 3.321, 6.093) C100 50 5.781 6.983 0.707 ( 4.395, 7.168)

Note that 96 of 100 confidence intervals contain the true value of the population meanµ = 5 .

Example 7A: The US department of Health collected data for 875 women, age 18 to 24. The sample

mean for their cholesterol level was 192 mg/100mL with a standard deviation of 40. Construct the

95% confidence interval for the mean of the population (i.e. the average cholesterol of all women age

18 to 24 in USA).

Solution: n x s= = =875 192 40, ,

This is clearly a large sample situation. So we use 2

En

Z sα= for the margin of error.

95% gives Z=1.96 and 2

1 9 406 2 7875

sE Z . .nα= = ≈

Thus x E± = ±192 2 7. or 189.3 < µ.< 194.7

Example 7B: The US department of Health collected data for 41men, age 18 to 24. The sample mean

for their LDL was 52 mg/100mL with a standard deviation of 10. Construct the 90% confidence

interval for the mean of the population (i.e. the average LDL of all men age 18 to 24 in USA).


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Example 8A: A random sample of size 36 student’s IQ is taken from an unknown population. Find

the 95% confidence interval for the population mean µ.

91 114 94 97 96 118 89 95 104 90 125 89 84 144

81 97 72 99 130 97 131 91 131 110 111 95 89 93

73 107 109 108 118 90 119 91.

Solution: From the data set in above we get n x s= = =36 102 17, , . Since σ is not given we

use2

En

t sα= to find the margin of error.

2

17 5 75166 5 83

2 036

sE .t . .nα= = = ≈ and

102 5 8x E .± = ± or 102-5.8 < µ <102+5.8 96.2 < µ <107.8.

Again we have a large sample and we can use 2

En

Z sα= for the margin of error. Note that

the distribution of the population is not known, but the sample size of 36 telling us we should

not be worried about the normality assumption. By CLT the distribution of the sample mean,

X , is normal.

As usual 95% confidence level gives Z=1.96 and 2

17 5 553 5 63

1 966

sE . .n

Z .α= = = ≈

So x E± = ±102 5 6. , 102-5.6=96.4, 102+5.6=107.6, 96.4 < µ< 107.6

As we observe the margin of error in the latter is a little smaller.

Example 8B: A random sample of size 16 student’s age is taken from a normally distributed

population. Find the 98% confidence interval for the population mean µ.

25 22 18 19 21 20 36 21

30 27 23 24 26 19 18 31


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Example 9: Use the data in example8 and construct the 90%, 92%, 96%, 98%, and 99% confidence

interval for the mean of the population:

Solution: From Example 8A we have n x s= = =36 102 17, , . We will use z and t for all cases.

Z Confidence Interval t Confidence Interval

80% Z=1.282 171 2823

3 636

E . .= ≈

98.4 < µ <105.6

t=1.306 171 3063

3 706

E . .= ≈

98.3 < µ <105.7

90% Z=1.645 176

4645 713

E ..= ≈

97.3 < µ <106.7

t=1.690 171 693

4 786

E . .= ≈

97.2 < µ <16.8

92% Z=1.75 113

57756

E .= ≈

97.0 < µ <107.0

t=1.803

171 8033

5 116

E . .= ≈

96.9 < µ <107.11

95% Z=1.645 171 6453

4 666

E . .= ≈

97.3 < µ <106.7

t=2.03 172 033

5 756

E . .= ≈

96.3 < µ <107.8

96% Z=2.05 172 0536

5 8.E .= ≈

96.4 < µ <107.5

t=2.13 172 133

6 046

E . .= ≈

995.96< µ <108.04

98% Z=2.33 172 3336

6 6.E .= ≈

95.4 < µ <108.6

t=2.44 172 443

6 916

E . .= ≈

95.1 < µ <108.9

99% Z=2.575 176

7575 323

E ..= ≈

94.7 < µ <109.3

t=2.724 172 5243

7 156

E . .= ≈

94.9 < µ <109.2

Note that the margin of error is getting larger as the confidence level is increasing and therefore

the confidence interval is getting wider as expected. Also notice that the margin of the error for z

and t are very close to each other.


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Example 10A: Suppose we have n x s= = =16 75 8, , from a normally distributed population with

unknown standard deviation. Construct the 95% confidence interval for the true mean of the

population, µ.

Solution: Clearly this is the small sample situation and σ is not known that means we must use t

distribution. So the margin of error is given by2

En

t sα= .

For 95%, we have α = =5% 0 05. with df =16-1=15 we get tα2

2 132= . (Use Table II)

2

8 4 264 4 31

2 1326

sE .t ..nα= = = ≈ . Hence x E± = ±75 4 3. and the 95% confidence

interval is given by 70 7 79 3. .µ< <

Example 10A: Suppose we have 16 75 8, ,n x σ= = = from a normally distributed population

with known standard deviation. Construct the 95% confidence interval for the true mean of the

population, µ.


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Example 11A: A random sample of size 20 data is taken from a normal population, given below. Find

the 90% confidence interval for the population mean µ.

55 65 58 96 74 85 52 63 87 78

54 57 65 55 85 75 76 74 70 72

Solution: Since σ is not known again the margin of error is given by2

En

t sα= . Use a

calculator or computer to verify 69 8 12 6x . , s .= =

2

12 6 4 81 729 71 4 8720

s .E . .n

t .α= = = ≈

Thus we have 69 80 4 87. .x E± = ± or 64 93 74 67. .µ< <

Example 11B: A random sample of size 20 data is taken from a normal population with known

standard deviation of the population 12 6.σ = , given below. Find the 90% confidence interval for the

population mean µ.

55 65 58 96 74 85 52 63 87 78

54 57 65 55 85 75 76 74 70 72


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Example 12A: The department of Bio-statistics wants to estimate the average age of students who

apply to their program with 95% confidence level. If they know the standard deviation of the ages is

known to be 8 years and the margin of error is 2 years, how large is the sample?

Solution: 95% level implies Zα2

1 96= . and E = 2 andσ = 8 . If we substitute in the above

formula, we get

2

2Z

nEα σ⎛ ⎞

⎜ ⎟=⎜ ⎟⎝ ⎠

21 96 8 622

. *⎛ ⎞= ≈⎜ ⎟⎝ ⎠

.

Example 12B: The department of mathematics wants to estimate the average GPA of students who

apply to their program with 99% confidence level. If they know the standard deviation of the GPA is

known to be 0.60 and the margin of error is 0.2, how large is the sample?

Example 13A: The school of nursing wants to estimate the average salary of their students at their

first job with 98% confidence level. A pilot study showed that the standard deviation of the population

is approximately $5,000. Also they want no more than $1,000 margin of error. How large sample is

necessary?

Solution: 98% level implies Zα2

2 33= . and E = 2,000 andσ = 5 000, . If we substitute in the above

formula, we get

2

2Z

nEα σ⎛ ⎞

⎜ ⎟=⎜ ⎟⎝ ⎠

22 33 5000 1361000

. *⎛ ⎞ ≈⎜ ⎟⎝ ⎠

.


20

Example 13B: A university dean wishes to estimate the percent his part-time instructors who teach at least 10 hours per week. How large a sample must be selected if he wants to be 97% confident of finding whether the true population proportion differs from the sample by no more that 5%?

Example 14A: A random sample of 200 students at UCG shows that 120 of them have internet access.

Find the 95% confidence interval for the population proportion of all students at UCG, who have

access to internet.

Solution: 95% implies Zα2

1 96= . , .p xn

= = =120200

0 60 , . .q p= − = − =1 1 0 60 0 40

2

0 60 0 401 0 0200

96 7.ˆ ˆpZ . .q *E .nα= = ≈ , . .p E± = ±0 60 0 07 .

Hence the 95% confidence interval is given by 0 53 0 67. .p< <

Example 14B: A random sample of 500 people in USA shows that 380 of them have cable TV. Find

the 95% confidence interval for the population proportion of all people in USA who have cable TV


21

Example 15A: A random sample of 500 students at UCG shows that 20% of them love STAT. Find

99% confidence interval for the population proportion of all students at UCG, who love STAT.


2 58= . . The sample proportion is given to be 20% .p = =20% 0 20 ,

. .q p= − = − =1 1 0 20 0 80

2

0 20 0 802 0 0500

58 5.ˆ ˆpZ . .q *E .nα= = ≈ , . .p E± = ±0 20 0 05 .

Hence the 99% confidence interval is given by 0 15 0 25. .p< < .

Example 15B: A random sample of 1280 students at CSULB shows that 80% of them are not

enjoying math courses. Find the 97% confidence interval for the population proportion of all students

at CSULB who are enjoying math classes.

Example 16A: A survey of 180 freshmen at CSUF showed that 135 of them like their professors that

are taking their courses for the first time. Find the 90% confidence interval for the proportion of all

students at CSUF who like their professors that are taking their courses for the first time.


1 645= . , .p = =135180

0 75 , . .q p= − = − =1 1 0 75 0 25

2

0 75 0 25 0 01 0

18

645 5.ˆ ˆpq * .n

Z .E .α= = ≈ , . .p E± = ±0 75 0 05 ,

Hence the 90% confidence interval is given by 0 70 0 80. .< <p .


22

Example 16B: A survey of 1201 women in Long Beach showed that 428 of them like their job very

much. Find the 99% confidence interval for the proportion of all women in Long Beach who like their

job very much.

Example 17A: School of Nursing wants to estimate, within 0.02, the true proportion of their students

who are passing STAT 414 with grade of B or better. They want to be 97% confident. Some study has

shown the proportion of students who receive B is about 20%.


2 17= . , .p = 0 20 and . .q p= − = − =1 1 0 20 0 80 with E=0.02

2

2Z

ˆ ˆn pqEα⎛ ⎞

⎜ ⎟=⎜ ⎟⎝ ⎠

22 170 20 0 80 18840 02.. * ..

⎛ ⎞= ≈⎜ ⎟⎝ ⎠

.

Example 17B: School of Medicine wants to estimate, within 3% of margin of errors, the true

proportion of their students who are passing Biology 522. They want to be 98% confident. The past

study shows the proportion of students who pass this course is about 70%.


23

Example 18A: Repeat example 17A for the case we have no prior information.

Solution: Remember that we there is no knowledge about the p, we use p = =50% 0 50.

So we have: .p = 0 50 , . .q p= − = − =1 1 0 50 0 50 , E=0.02

2

2Z

ˆ ˆn pqEα⎛ ⎞

⎜ ⎟=⎜ ⎟⎝ ⎠

22 170 50 0 50 29440 02.. * ..

⎛ ⎞= ≈⎜ ⎟⎝ ⎠

Example 18B: Repeat example 17B for the case we have no prior information.

Example 19A: School of Nursing wants to estimate, within 0.02, the true proportion of their students



Example 19B: School of Nursing wants to estimate, within 0.02, the true proportion of their students




24

Example 20A: The cholesterol concentration in the yolks of each of a sample of 18 randomly selected

eggs laid by genetically engineered chickens were found to have a sample mean value, of 9.38 mg/g of

yolk and a standard deviation, s, of 1.62 mg/g. Construct a 95% confidence interval estimate of the true

variance and standard deviation of the cholesterol concentration in these egg yolks.

Solution:

2

2αχ : 2 2

0 05 0 0252

30 191. . .χ χ == from Table III,

21 2αχ−

: 2 2 20 05 1 0 025 0 9751 2

7 564. . . .χ χ χ−−= = =

( ) ( )2 22

2 212 2

1 1n s n s

α αχ χσ

−

− −< <

2 22

30 191 7 517 1 62

6624

17 1. .

. .σ× ×⇒ ≤ ≤

Variance: 1.478 ≤ σ2 ≤ 5.898

21 478 5 898. .σ≤ ≤

Standard Deviation: 1.216 ≤ σ ≤ 2.429 Example 20B: A random sample of size 20 data is taken from a normal population, given below. Find

the 90% confidence interval for the population variance and standard deviation.

75 55 68 86 94 65 92 73 78 98

64 67 55 95 75 85 86 64 70 82


25

Example 21A: A random sample of 64 students at UCG yields x = 2 94. and s = 0 60. . Construct the

95% confidence interval for the true population standard deviationσ .

Solution:

2

2αχ : 2 2

0 05 0 0252

83 298. . .χ χ == from Table III,

21 2αχ−

: 2 2 20 05 1 0 025 0 9751 2

40 482. . . .χ χ χ−−= = =

( ) ( )2 22

2 212 2

1 1n s n s

α αχ χσ

−

− −< < ( ) ( )2 2

2

83 2964

8 40 41 0 60 64

821 0 60

. .. .

σ− −

⇒ < < ,

63 0 3683 298

63 0 3640 482

2b g b g..

..

< <σ

0 261 0 5282. .< <σ .

Hence the 95% confidence interval for the standard deviation of the population is given by

0 51 0 73. .σ< < .

Example 21B: Suppose we have 16 75 8, ,n x s= = = from a normally distributed population

with unknown standard deviation. Construct the 95% confidence interval for the true population

standard deviationσ .


26

Example 22A: A random sample of size 31 data is taken from a normal population, given below.

55 65 58 96 74 85 52 63 87 78 96 98 54

54 57 65 55 85 75 76 74 70 72 86 78 64

68 78 88 98 58

A- Find the 90% confidence interval for the population mean µ.

B- Find the 90% confidence interval for the population standard deviationσ .

Example 22B: A random sample of size 12 data is taken from a normal population, given below.

55 65 58 96 74 68

54 57 65 55 85 78

A- Find the 90% confidence interval for the population mean µ.

B- Find the 90% confidence interval for the population standard deviationσ .


27

Example 23A- A random sample of 30 of the paid attendances is shown. Find the 95% confidence interval for the mean paid attendance at the Major League All Star games. . 47,596 68,751 50,838 31,938 31,851 56,088 69,831 28,843 53,107 34,906 38,359 72,086 31,391 48,829 50,706 34,009 50,850 43,801 62,892 55,105 63,974 46,127 49,926 54,960 56,674 38,362 51,549 32,785 48,321 49,671 Example 23B- A random sample of 20 of students who have taken the test #1 from the past many STAT1 tests is shown. Find the 95% confidence interval for the mean of all students who have taken test #1 in the past in STAT1. It is assumed the population of all exams in STAT1 is normally distributed.

59 75 54 69 58 61 75 71 79 84 49 67 72 70 77 61 75 70 55 65


28

Example 24A-A sample of the reading scores of 35 fifth-graders has a mean of 82. The standard

deviation of the sample is 15. Assume the population is normally distributed.

A-Find the 95% confidence interval of the mean reading scores of all fifth-graders.

B-Find the 95% confidence interval of the standard deviation of the reading scores of all fifth-graders.

Example 24B-A study found that 8 to 12 year-olds spend an average of $18.50 per trip to a mall. If a sample of 49 children was used. Assume the standard deviation of the sample is $1.56. A- Find the 98% confidence interval of the mean of the population. B- Find the 98% confidence interval of the standard deviation of the population.


29

Example 25A-Noise levels at various area urban hospitals were measured in decibels. The mean of the noise levels in 84 corridors was 61.2 decibels, and the standard deviation was 7.9. Find the 95% confidence interval of the true mean.

Example 25B- It is claimed that the distribution of monthly parking fees in a city is normal with a

mean of $65.00. In order to test the truth of this claim, you randomly sampled 26 residents in the city

and asked them how much they pay for parking each month. The mean was $62.50 with a standard

deviation of $5.00. Find the 95% confidence interval of the true mean and check to see if the claim is

correct.


30

Example 26A-A university dean wishes to estimate the average number of hours his part-time

instructors teach per week. The standard deviation from a previous study is 2.6 hours. How large a

sample must be selected if he wants to be 99% confident of finding whether the true mean differs from

the sample mean by 1 hour? n = 45

Example 26B-An insurance company is trying to estimate the average number of sick days that full-

time food-service workers use per year. A pilot study found the standard deviation to be 2.5 days.

How large a sample must be selected if the company wants to be 95% confident of getting an interval

that contains the true mean with a maximum error of .5 day? n = 100

Example 26C-A health care professional wishes to estimate the birth weight of infants. How large a

sample must she select if she desires to be 90% confident that the true mean is within 2 ounces of the

sample mean? The standard deviation of the birth weights is known to be 10 ounces. n = 68


31

Example 27A-A university dean wishes to estimate the percent his part-time instructors who teach at

least 10 hours per week. How large a sample must be selected if he wants to be 97% confident of

finding whether the true population proportion differs from the sample by no more that 5%?n = 471

Example 27B-An insurance company is trying to estimate the proportion of sick days that full-time

food-service workers use per year. A pilot study found this proportion is about 25%. How large a

sample must be selected if the company wants to be 95% confident of getting an interval that contains

the true proportion with margin of error of 6 percentage point? n = 201

Example 27C-A health care professional wishes to estimate the percentage of over weight infants.

How large a sample must she select if she desires to be 90% confident that the true proportion is

within 2 percentage of the population proportion? n = 1692


32

Example 28A-A sample of 6 adult elephants had an average weight of 12,200 pounds, with a sample

standard deviation of 200 pounds. Find the 95% confidence interval of the true mean. The data comes

from a normally distributed population.

Example 28B-A recent study of 38 city residents showed that the mean of the time they had lived at

their present address was 9.3 years. The standard deviation of the sample was 2 years. Find the 90%

confidence interval of the true mean.

Example 28C-A recent study of 25 students showed that they spent an average of $18.53 for gasoline

per week. The standard deviation of the sample was $3.00. Find the 95% confidence interval of the

true mean.


33

Example 29A- A U.S Travel Data Center survey conducted for Better Homes and Gardens of 1500

adults found that 39% said that they would take more vacations this year than last year. Find the 95%

confidence interval of the true proportion of adults who said that they will travel more this year.

Example 29B-An employment counselor found that in a sample of 100 unemployed workers, 65%

were not interested in returning to work. Find the 95% confidence interval of the true proportion of

workers who do not wish to return to work.

Example 29C- A survey found that out of 200 workers, 168 said they were interrupted three or more

times an hour by phone messages, faxes, etc. Find the 90% confidence interval of the population

proportion of workers who are interrupted three or more times per hour.


34

Example 30A-A study by the University of Michigan found that one in five 13- and 14- year-old is a

sometime smoker. To see how the smoking rate of students at a large school district compared to the

national rate, the superintendent surveyed 200 13- and 14- year-old students and found that 23% said

they were sometime smokers. Find the 99% confidence interval of the true proportion and compare

this with the University of Michigan's study.

Example 30B-A survey of 90 families showed that 40 owned at least one gun. Find the 95%

confidence interval of the true proportion of families who own at least one gun.

Example 30C-A survey of 800 students showed that 60% have smoked at least once in their life. Find

the 95% confidence interval of the true proportion of students who have smoked at least once in their

life.


35

Gegative Z-Score 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

-3.80 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.70 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.60 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.50 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 -3.40 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 -3.30 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 -3.20 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 -3.10 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 -3.00 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 -2.90 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 -2.80 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 -2.70 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 -2.60 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 -2.50 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 -2.40 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 -2.30 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 -2.20 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 -2.10 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 -2.00 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 -1.90 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 -1.80 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 -1.70 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 -1.60 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 -1.50 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 -1.40 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 -1.30 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 -1.20 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 -1.10 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 -1.00 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 -0.90 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 -0.80 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 -0.70 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 -0.60 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 -0.50 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 -0.40 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 -0.30 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 -0.20 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 -0.10 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 0.00 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

All Tables are derived by Dr. Ghamsary


36

Positive Z-Score

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 4.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000


37

Student t-Distribution Table II CI-Level 80% 90% 95% 98% 99%

Two Tails 0.20 0.10 0.05 0.02 0.01 One Tail 0.1 0.05 0.025 0.01 0.005

1 3.078 6.314 12.706 31.821 63.657 2 1.886 2.920 4.303 6.965 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.533 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032 6 1.440 1.943 2.447 3.143 3.707 7 1.415 1.895 2.365 2.998 3.499 8 1.397 1.860 2.306 2.896 3.355 9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169 11 1.363 1.796 2.201 2.718 3.106 12 1.356 1.782 2.179 2.681 3.055 13 1.350 1.771 2.160 2.650 3.012 14 1.345 1.761 2.145 2.624 2.977 15 1.341 1.753 2.131 2.602 2.947 16 1.337 1.746 2.120 2.583 2.921 17 1.333 1.740 2.110 2.567 2.898 18 1.330 1.734 2.101 2.552 2.878 19 1.328 1.729 2.093 2.539 2.861 20 1.325 1.725 2.086 2.528 2.845 21 1.323 1.721 2.080 2.518 2.831 22 1.321 1.717 2.074 2.508 2.819 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 26 1.315 1.706 2.056 2.479 2.779 27 1.314 1.703 2.052 2.473 2.771 28 1.313 1.701 2.048 2.467 2.763 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750 31 1.309 1.696 2.040 2.453 2.744 32 1.309 1.694 2.037 2.449 2.738 33 1.308 1.692 2.035 2.445 2.733 34 1.307 1.691 2.032 2.441 2.728 35 1.306 1.690 2.030 2.438 2.724 40 1.303 1.684 2.021 2.423 2.704 50 1.299 1.676 2.009 2.403 2.678 60 1.296 1.671 2.000 2.390 2.660 70 1.294 1.667 1.994 2.381 2.648 80 1.292 1.664 1.990 2.374 2.639 90 1.291 1.662 1.987 2.368 2.632 100 1.290 1.660 1.984 2.364 2.626

large 1.282 1.645 1.960 2.326 2.576


38

Chi-Square Table III

Prob 0.005 0.01 0.025 0.05 0.1 0.9 0.95 0.975 0.99 0.995 1 7.844 6.635 5.024 3.841 2.706 0.016 0.004 0.001 0.000 0.000 2 10.557 9.210 7.378 5.991 4.605 0.211 0.103 0.051 0.020 0.010 3 12.796 11.345 9.348 7.815 6.251 0.584 0.352 0.216 0.115 0.072 4 14.815 13.277 11.143 9.488 7.779 1.064 0.711 0.484 0.297 0.207 5 16.702 15.086 12.833 11.070 9.236 1.610 1.145 0.831 0.554 0.412 6 18.499 16.812 14.449 12.592 10.645 2.204 1.635 1.237 0.872 0.676 7 20.227 18.475 16.013 14.067 12.017 2.833 2.167 1.690 1.239 0.989 8 21.902 20.090 17.535 15.507 13.362 3.490 2.733 2.180 1.646 1.344 9 23.535 21.666 19.023 16.919 14.684 4.168 3.325 2.700 2.088 1.735

10 25.132 23.209 20.483 18.307 15.987 4.865 3.940 3.247 2.558 2.156 11 26.700 24.725 21.920 19.675 17.275 5.578 4.575 3.816 3.053 2.603 12 28.241 26.217 23.337 21.026 18.549 6.304 5.226 4.404 3.571 3.074 13 29.760 27.688 24.736 22.362 19.812 7.042 5.892 5.009 4.107 3.565 14 31.258 29.141 26.119 23.685 21.064 7.790 6.571 5.629 4.660 4.075 15 32.739 30.578 27.488 24.996 22.307 8.547 7.261 6.262 5.229 4.601 16 34.204 32.000 28.845 26.296 23.542 9.312 7.962 6.908 5.812 5.142 17 35.654 33.409 30.191 27.587 24.769 10.085 8.672 7.564 6.408 5.697 18 37.091 34.805 31.526 28.869 25.989 10.865 9.390 8.231 7.015 6.265 19 38.515 36.191 32.852 30.144 27.204 11.651 10.117 8.907 7.633 6.844 20 39.929 37.566 34.170 31.410 28.412 12.443 10.851 9.591 8.260 7.434 21 41.332 38.932 35.479 32.671 29.615 13.240 11.591 10.283 8.897 8.034 22 42.726 40.289 36.781 33.924 30.813 14.041 12.338 10.982 9.542 8.643 23 44.110 41.638 38.076 35.172 32.007 14.848 13.091 11.689 10.196 9.260 24 45.486 42.980 39.364 36.415 33.196 15.659 13.848 12.401 10.856 9.886 25 46.855 44.314 40.646 37.652 34.382 16.473 14.611 13.120 11.524 10.520 26 48.216 45.642 41.923 38.885 35.563 17.292 15.379 13.844 12.198 11.160 27 49.570 46.963 43.195 40.113 36.741 18.114 16.151 14.573 12.879 11.808 28 50.918 48.278 44.461 41.337 37.916 18.939 16.928 15.308 13.565 12.461 29 52.259 49.588 45.722 42.557 39.087 19.768 17.708 16.047 14.256 13.121 30 53.594 50.892 46.979 43.773 40.256 20.599 18.493 16.791 14.953 13.787 31 54.924 52.191 48.232 44.985 41.422 21.434 19.281 17.539 15.655 14.458 32 56.249 53.486 49.480 46.194 42.585 22.271 20.072 18.291 16.362 15.134 33 57.568 54.776 50.725 47.400 43.745 23.110 20.867 19.047 17.074 15.815 34 58.883 56.061 51.966 48.602 44.903 23.952 21.664 19.806 17.789 16.501 35 60.193 57.342 53.203 49.802 46.059 24.797 22.465 20.569 18.509 17.192 36 61.499 58.619 54.437 50.998 47.212 25.643 23.269 21.336 19.233 17.887 37 62.800 59.893 55.668 52.192 48.363 26.492 24.075 22.106 19.960 18.586 38 64.097 61.162 56.896 53.384 49.513 27.343 24.884 22.878 20.691 19.289 39 65.391 62.428 58.120 54.572 50.660 28.196 25.695 23.654 21.426 19.996 40 66.680 63.691 59.342 55.758 51.805 29.051 26.509 24.433 22.164 20.707 50 79.397 76.154 71.420 67.505 63.167 37.689 34.764 32.357 29.707 27.991 60 91.852 88.379 83.298 79.082 74.397 46.459 43.188 40.482 37.485 35.534 70 104.110 100.425 95.023 90.531 85.527 55.329 51.739 48.758 45.442 43.275 80 116.210 112.329 106.629 101.879 96.578 64.278 60.391 57.153 53.540 51.172 90 128.183 124.116 118.136 113.145 107.565 73.291 69.126 65.647 61.754 59.196

100 140.048 135.807 129.561 124.342 118.498 82.358 77.929 74.222 70.065 67.328


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elementary statisticselementary statistics chapter 06 dr. ghamsary page 5 5 sample size the...

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