1. Homework #22. Inferential Statistics 3. Review for Exam

HOMEWORK #2: Part A Sanitation Eng. Z=.53 = .2019 + .50 = .7019 F.C. Z=.67 = .2486 + .50 = .7486

5 GPA’s, which are in the top 10%? GPA of 3.0 and 3.20 are not:

Z = (3.0-2.78)/.33 =.67 Area beyond = .2514 (25.14%)

Z=(3.20-2.78)/.33=1.27 Corresponds to .8980 (.3980+.5000) Area beyond = .1020 (10.2%)

By contrast, for 3.21… Z=(3.21-2.78)/.33=1.30

Corresponds to .9032 (.4032+.5000)

HOMEWORK #2: Part B

Question 1 a. Mean=18.87; median=15; mode=4

b. The mean is higher because the distribution is positively skewed (several large cities with high percents)

c. When you remove NYC, the mean=16.43 & the median goes from 15 to 14.5. Removing NYC’s high value from the distribution reduces the skew. The mean decreases more than the median because value

of the mean is influenced by outlying values; the median is not—it only moves one case over.

HOMEWORK #2: Part B Question 2

For this problem, there are two measures of central tendency (indicating the “typical” score). The mean per student expenditure was almost $2,000 higher in

2003 ($9,009) than in 1993 ($7,050). The median also increased, but not nearly as much (from $7,215

to $7,516).

The spread of the scores, as indicated by the standard deviation, was more than double 2003 (1,960) than it was in 1993 (804).

Shape For 1993, the distribution of scores has a slight negative skew;

this distribution is essentially normal (bell-shaped) as the mean ($7,050) and median ($7,215) are similar. By contrast, for 2003, the mean is much greater than the median; this distribution has a strong positive skew.

HOMEWORK #2: Part B Q3

a. 53.28% Opposite sides of mean, add 2 areas together

b. 6.38% Both scores on right side of mean, subtract areas

c. 10.56% “Column C” area for Z=1.25 is .1056

d. 69.15% “Column B” area for Z= -0.5 is .1915 + .5000 (for other half

of normal curve) e. 99.38%

Z=2.5; Column B (for area between 2.5 & 0) = .4938 + .5000 (for other half of normal curve)

f. 6.68% Z = -1.5; Column C for area beyond -1.5 =.0668

HOMEWORK #2: Part B Q4

a. .9953 Column B area (.4953) + .5000 (for other half of normal

curve) b. .5000

50% of area on either side of mean (47) c. .6826

“Column B” for both – .3413 + .3413 d. .9997

Column B area (.4997) + .5000 (for other half of normal curve)

e. .0548 “Column C” area for Z=1.6

f. .3811 Scores on opposite sides of mean add “Col. B” areas

HOMEWORK #2: Part C

SPSS: All the info needed to answer these questions

is contained in this output

StatisticsHOURS PER DAY WATCHING TVN Valid 1426

Missing 618Mean 3.03Median 2.00Mode 2Std. Deviation 2.766Percentiles 10 1.00

20 1.0025 1.0030 2.0040 2.0050 2.0060 3.0070 3.0075 4.0080 4.0090 6.00

Distribution (Histogram) for TV Hours

Sibs Distribution

College Science Credits

Sampling Terminology Element: the unit of which a population is

comprised and which is selected in the sample Population: the theoretically specified

aggregation of the elements in the study (e.g., all elements)

Parameter: Description of a variable in the population σ = standard deviation, µ = mean

Sample: The aggregate of all elements taken from the pop.

Statistic: Description of a variable in the sample (estimate of parameter) X = mean, s = standard deviation

Non-probability Sampling

Elements have unknown odds of selection Examples

Snowballing, available subjects… Limits/problems

Cannot generalize to population of interest (doesn’t adequately represent the population (bias)

Have no idea how biased your sample is, or how close you are to the population of interest

Probability Sampling

Definition: Elements in the population have a known (usually

equal) probability of selection Benefits of Probability Sampling

Avoid bias Both conscious and unconscious More representative of population

Use probability theory to: Estimate sampling error Calculate confidence intervals

Sampling Distributions

Link between sample and population DEFINITION 1

IF a large (infinite) number of independent, random samples are drawn from a population, and a statistic is plotted from each sample….

DEFINITION 2 The theoretical, probabilistic distribution of a

statistic for all possible samples of a certain outcome

The Central Limit Theorem I

IF REPEATED random samples are drawn from the population, the sampling distribution will always be normally distributed As long as N is sufficiently (>100) large

The mean of the sampling distribution will equal the mean of the population WHY? Because the most common sample mean will

be the population mean Other common sample means will cluster around the

population mean (near misses) and so forth Some “weird” sample findings, though rare

The Central Limit Theorem II

Again, WITH REPEATED RANDOM SAMPLES, The Standard Deviation of the Sampling distribution = σ

√N This Critter (the population standard deviation

divided by the square root of N) is “The Standard Error” How far the “typical” sample statistic falls from the

true population parameter

The KICKER

Because the sampling distribution is normally distributed….Probability theory dictates the percentage of sample statistics that will fall within one standard error

1 standard error = 34%, or +/- 1 standard error = 68% 1.96 standard errors = 95% 2.58 standard errors = 99%

The REAL KICKER

From what happens (probability theory) with an infinite # of samples… To making a judgment about the accuracy of statistics

generated from a single sample Any statistic generated from a single random sample has a

68% chance of falling within one standard error of the population parameter OR roughly a 95% CHANCE OF FALLING WITHIN 2 STANDARD

ERRORS

EXAM Closed book

BRING CALCULATOR

You will have full class to complete

Format: Output interpretation Z-score calculation problems

Memorize Z formula Z-score area table provided

Short Answer/Scenarios Multiple choice

Review for Exam Variables vs. values/attributes/scores

variable – trait that can change values from case to case example: GPA

score (attribute)– an individual case’s value for a given variable

Concepts Operationalize Variables

Review for Exam Short-answer questions, examples:

What is a strength of the standard deviation over other measures of dispersion?

Multiple choice question examples: Professor Pinhead has an ordinal measure of a variable called

“religiousness.” He wants to describe how the typical survey respondent scored on this variable. He should report the ____. a. median b. mean c. mode e. standard deviation

On all normal curves the area between the mean and +/- 2 standard deviations will be a. about 50% of the total area b. about 68% of the total area c. about 95% of the total area d. more than 99% of the total area

EXAM

Covers chapters 1- (part of)6: Chapter 1

Levels of measurement (nominal, ordinal, I-R) Any I-R variable could be transformed into an ordinal or

nominal-level variable Don’t worry about discrete-continuous distinction

Chapter 2 Percentages, proportions, rates & ratios

Review HW’s to make sure you’re comfortable interpreting tables

EXAM Chapter 3: Central tendency

ID-ing the “typical” case in a distribution Mean, median, mode

Appropriate for which levels of measurement? Identifying skew/direction of skew Skew vs. outliers

Chapter 4: Spread of a distribution R & Q s2 – variance (mean of squared deviations) s

Uses every score in the distribution Gives the typical deviation of the scores

DON’T need to know IQV (section 4.2)

Keep in mind…

All measures of central tendency try to describe the “typical case” Preference is given to statistics that use the most

information For interval-ratio variables, unless you have a highly

skewed distribution, mean is the most appropriate For ordinal, the median is preferred

If mean is not appropriate, neither is “s” S = how far cases typically fall from mean

EXAM Chapter 5

Characteristics of the normal curve Know areas under the curve (Figure 5.3)

KNOW Z score formula Be able to apply Z scores

Finding areas under curve Z scores & probability Frequency tables & probability

EXAM Chapter 6

Reasons for sampling Advantages of probability sampling What does it mean for a sample to be representative? Definition of probability (random) sampling Sampling error

Plus… Types of nonprobability sampling

Interpret Total IQ Score N Valid 1826

Missing 9092

Mean 88.98

Median 91.00

Mode 94

Std. Deviation 20.063

Minimum 0

Maximum 160

Percentiles 10 63.00

20 74.00

25 78.00

30 80.00

40 86.00

50 91.00

60 95.00

70 100.00

75 103.00

80 105.00

90 112.00

1. Number of cases used to calculate mean?

2. Most common IQ score?

3. Distribution skewed? Direction?

4. Q?5. Range?6. Is standard

deviation appropriate to use here?

Scenario

Professor Scully believes income is a good predictor of the size of a persons’ house IV? DV? Operationalize DV so that it is measured at all

three levels (nominal, ordinal, IR) Repeat for IV

Express the answer in the proper format

Percent Proportion Ratio Probability