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CHAPTER 8Homework:1,3,7,17,23,26,31,33,39,41,47,49,55,61,67,69,100,1

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Sec 8.1: Elements of a hypothesis Testing:

(1) Set up hypotheses A hypothesis is simply a statement about a population parameter, e.g. the population mean.

There are two types of hypotheses -- the null hypothesis and alternative hypothesis.

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A NULL HYPOTHESIS is a hypothesis to be tested. Typically, we believe that null-hypothesis is true unless the data provide enough evident that it is false.

AN ALTERNATIVE HYPOTHESIS is a hypothesis that contradicts the null-hypothesis. Ifthe null hypothesis is rejected by a test, then we believe the alternative hypothesis is true.

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Remember: If the null hypothesis is not rejected by a test, we can not infer that the null hypothesis is true. That is, a hypotheses test can only prove (with a confidence) that the alternative may be true, but never the null.

Because of this special feature of a hypotheses testing procedure, the alternative hypothesis is usually set up as a hypothesis that is hoped to be shown to be true by the test.

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TWO-TAILED ALTERNATIVEIf the alternative states that a population parameter is different from a specific value. The corresponding test is called a two-tailed test.

RIGHT-TAILED ALTERNATIVEIf the alternative states that a population parameter is greater than a specific value. The corresponding test is called a right-tailed test.

LEFT-TAILED ALTERNATIVEIf the alternative states that a population parameter is less than a specific value. The corresponding test is called a left-tailed test.

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(EX 8.1) (Basic-- Set up the hypotheses)The R.R Bowker company of New-York

collects information on the retail prices of books. In 1986, the mean retail price of all hardcover history books was $28.44. Suppose you want to know whether the mean retail price of this kind of books is higher than $28.44 this year. Can you set up a test answering your problem?(a). Determine the alternative hypothesis.(b). Determine the null hypothesis.(c). What type of hypothesis it is?

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(EX 8.2) (Basic -- Set up the hypotheses)High airline occupancy rates on scheduled

flights are essentially to profitability. Suppose that a scheduled flight must average at least 60% occupancy rate to be profitable. We know that the occupancy rate of the Sunday morning flight from Orlando to New-York City is only 54%. Before the company decides to close this scheduled flight, they ask you to set up a test helping them to make their decision. (a). What should be the alternative hypothesis if the company's goal is to close this flight?(b). What is the null hypothesis?

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(2) Compute the test statistic

We already discussed that there are several different types of statistics to measure the central tendency of a population. Also, there are several different test statistics for testing about a population mean.

For example, there are z-statistic and t-statistic.

Which one should be used depends on assumptions requied by these tests, as in the construction of

confidence intervals.

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(3) Decide the rejection region of the test Based on the test statistic and a given confidence level, we can determine the rejection region, the acceptance region, and the critical value of the test.

Rejection region is the region in which we can reject the null-hypothesis when the test statistics falls in this region. Acceptance region is simply the complement of the rejection region.

Critical value is the value (or values) on the boundary of the rejection region and acceptance region.

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(4) p-value and hypotheses testing As an alternative approach to the

rejection/acceptance-region approach, we can calculate a probability related to the test statistic, called P-value, and base our decision of rejection/acceptance on the magnitude of the P-value.

P-value is the probability to observe a value of the test statistic as extreme as the one observed, if the null hypothesis is true. So a small P-value indicates that the null hypothesis is not true and hence should be rejected.

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(5) Two possible errors in hypotheses testing, and the size/significance level of a test There are two types of error which will occur in a statistical test of hypotheses.

Type I error occurs when you reject a null-hypothesis while it is true.

Type II error occurs when you fail to reject a false null-hypothesis.

The probability of making type I error is called the size or significant level () of the test, often denoted as .

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Sec 8.2 Large Sample Test for a population mean

For a large sample, usually the sample size > 30, the central limiting theorem ensures that the sample mean is at least approximately normally distributed for a wide range of sampled populations. Also, the sample variance provides a good estimation for the unknown population variance. Therefore, we can use the standard normal z test statistic to complete our test.

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Large sample test for a population mean

(a) Alternative Hypothesis:(i) Two-Tailed Test: Ha: (ii) Right-Tailed Test: Ha:

(iii) Left-Tailed Test: Ha:

(b) Null Hypothesis:(i) Two-Tailed Test: Ha: (ii) Right-Tailed Test: Ha:

(iii) Left-Tailed Test: Ha:

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(c) Test Statistic

If the population standard deviation is unknown, we can use the sample standard deviation to replace it, i.e.

Zc x

x

0

Zc xsx

0

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(d) Rejection Region of the test

If it is required that the size of the test is , then the rejection region is given by

(i) Two-Tailed Test: z > Z or z < -Z, (ii) Right-Tailed Test: z > Z , (iii) Left-Tailed Test: z < -Z.

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(e) P-Value of this test

(i) Two-Tailed Test: P-value = 2 * P(z > |Zc|), (ii) Right-Tailed Test: P-value = P (z > Zc), (iii) Left-Tailed Test: P-value = P(z < Zc).

If it is required that the size of the test should be then the null hypothesis is rejected if and only if the P-value is smaller than

The conclusion, either rejection or acceptance, of this procedure is exactly the same as the test based on the rejection region in (d).

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(EX 8.3) (Basic)A sample of n=35 observations from a long

tail population produced a mean equal to 2.4 and standard deviation equal to 0.29. Suppose that your research project is to show that the population mean exceeds 2.3.(a). Give the null and the alternative hypotheses of the test.(b). Find the test statistics and the p-value.(c). State the assumption you need.(d). Locate the rejection region of this test at 0.05 level and make your decision at 0.05 level.(e). Describe what types of error are possible in this decision process.

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(EX 8.4) (Basic)Refer to example 8.3. Suppose that your

research goal is to show that the population mean is less than 2.9.(a). Give the null and alternative hypotheses of the test.(b). Locate the rejection region of this test at 0.05 level.(c). Make your decision. (d). Describe possible erros in your decision.

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(EX 8.5) (Basic)Refer to example 8.3. Suppose that your

research goal is to show that the population mean differs from 2.45.(a). Give the null and alternative hypotheses of the test.(b). Locate the rejection region of this test at 0.05 level.(c). Make your decision. (d). Describe the types of error possible in this decision process.

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(EX 8.6) (Intermediate)A drug manufacturer claimed that the mean

potency of one of its antibiotics was 0.8. A sample of n = 100 capsules were tested and produced a sample mean equal to 0.797 with a standard deviation equal to 0.008. Do the data present sufficient evident to refute the manufacture's claim? (a). Give the null and alternative hypotheses of the test and find the test statistics.(b). State the assumption you need.(c). Find the p-value of this test and locate the rejection region of this test at 0.05 level.(d). Make your decision and describe what type of error possible in your decision.

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Sec 8.3: Small Sample Test for one

Population Mean

If we can assume the population we are interested in has a normal distribution, then we test the hypotheses using the t statistic, irrespective the size of the sample (whether it is small or large).

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(a). Alternative Hypothesis:(i) Two-Tailed Test: Ha: (ii) Right-Tailed Test: Ha:

(iii) Left-Tailed Test: Ha:

(b) Null-Hypothesis:(i) Two-Tailed Test: Ha: (ii) Right-Tailed Test: Ha:

(iii) Left-Tailed Test: Ha:

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(c) Test Statistics

tc xSx

0 .

(d) Rejection Region of the testA size test has the following rejection region:

(i). Two-Tailed Test: t > tn-1 or t < -tn-1, (ii). Right-Tailed Test: t > t,n-1, (iii). Left-Tailed Test: t < -t,n-1.

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(e). P-Value of this test:

(i). Two-Tailed Test: P-value=2*P( t > |tc |)(ii).Right-Tailed Test: P-value = P(t > tc)(iii).Left-Tailed Test: P-value = P(t < tc)

The null hypothesis is rejected if and only if the P-

value is less than , and this test reaches the same conclusion and the test based on the rejection region in (d).

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(EX 8.7) (Basic)The test statistics for testing a right-tailed test

with a sample of n=15 observations has the value tc=1.82.

(a). State the assumptions you need. What are the degrees of freedom for this statistics?

(b). Find the p-value of this test.(c). Give the rejection region of the test at 0.05

level and make your decision.(d). Give the rejection region of the test at

0.01 level and make your decision.(e). Describe what types of errors can possibly

be made in (c) and (d).

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(EX 8.8) (Basic)A manufacturer of gunpowder has

developed a new powder that is designed to produce a muzzle velocity of 3000 feet per second. Eight shells are loaded with the charge and the muzzle velocities measured. The resulting velocities are shown in the following table. Does this set of data provide enough information to claim that the muzzle velocity are less than 3000.Muzzle Velocities(feet per second)3005 2925 2995 29353005 2965 2935 2905

Note: x and s 2958 75 39 26. .

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(a). Give the null and alternative hypotheses of the test.(b). Find the test statistics.(c). State the assumptions you need.(d). Find the p-value of this test.(e). Locate the rejection region of this test at 0.05 level.(f). Make your decision.(g). Describe what types of error are possible in this type of decision.

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(EX 8.9) (Applications)"Lake Champlain Found to be Polluted by

PCBs," reports the New York Times(June 16, 1985). PCBs, a group of chemicals used for years as an insulator in some electrical equipment, have been found to cause cancer in laboratory animals and are suspected of similar effects on humans. Although the federal level of tolerance of PCBs in fish is two PPM, a sampling of 15 American eels in Lake Champlain gave PCB readings ranging from 4.05 to 19.49 PPM with a mean value and standard deviation of 9.84 and 3.86, respectively.

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(a). Give the null and alternative hypotheses of the test.(b). Find the test statistics.(c). State the assumptions you need.(d). Find the p-value of this test.(e). Locate the rejection region of this test at 0.10 level.(f). Make your decision.(g). Describe what types of error can possible be made in this type of decision.

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Sec 8.4 Large Sample Test for

a population proportion

The properties of a sample proportion were discussed in the previous chapter. For a sufficiently large sample, the sampling distribution of the sample proportion is approximately normal.

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(a). Alternative hypothesis

(i) Two-Tailed Test: Ha: p p(ii) Right-Tailed Test: Ha: p p

(iii) Left-Tailed Test: Ha: p p

(b) Null Hypothesis:

(i) Two-Tailed Test: Ha: pp(ii) Right-Tailed Test: Ha: p p

(iii) Left-Tailed Test: Ha: p p

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(c) Test Statistics:

Under the assumption that the null hypothesis is true, the population standard deviation should be estimated by

Zc p p

p

0

0

pp p

n00 01

( )

.

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(d) Assumption The sample size is large enough, i.e. the interval

must be contained in the interval (0 , 1), so that the sampling distribution of the test statistic can be approximated by a normal distribution .

(p0 3 30 0 0* , * ) p pp

(e) Reject Region of the test A size test has the following rejection region

(i). Two-Tailed Test: z < -Zor z > Z , (ii). Right-Tailed Test: z > Z, (iii). Left-Tailed Test: z < -Z.

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(f) P-Value of this test

(i). Two-Tailed Test: P-value=2*P(z > |Zc|)(ii). Right-Tailed Test: P-value=P(z > Zc)(iii). Left-Tailed Test: P-value=P(z < Zc)

A size test rejects the null hypothesis if and only if the p-value is less than . And the conclusion of this test is the same as the test in (d).

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(EX 8.10) (Basic)Regardless of age, about 20% of American

adults participate in fitness activities at least twice a week. However, the fitness activities are different among the students in UCF. In a local survey of n=100 students randomly selected from UCF, a total of 27 students indicated that they participated in a fitness activity at least twice a week. Does this data indicates that the UCF students’ participation rate differs significantly from the 20% national average at= 0.10?

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(EX 8.11) (Basic -- Large sample test)

A random sample of n = 1000 observations from a binomial population produced x = 279.(a). If your research hypothesis is that p is less than 0.3, what should you choose for your alternative hypothesis and null hypothesis?(b). Does your alternative hypothesis in part (a) imply a one or two tailed test? Explain.(c). Find the test statistics.(d). Does the data set provide sufficient evidence to indicate that p is less than 0.3 at= 0.05?

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(EX 8.12) (Applications)

More than ever before, Americans are working at two jobs, according to a Labor Department survey reported in the Wall Street Journal (November 7, 1994). According to the survey, the proportion of employed Americans holding two or more jobs is 7.2% compared to 6.2% in 1989. Assume that the current survey was based on a random sample of 950 employed Americans. If you wish to show that the proportion of Americans holding two or more jobs is greater than the 1989 figure,

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(a). State the null and alternative hypotheses to be tested.

(b). Locate the rejection region for 0.01 level.(c). Conduct the test and state your conclusion.