Chapter 10
Statistical Inference About Means and Proportions
with Two Populations

Case Problem: Par, Inc.

This case can provide discussion and differing opinions as to what hypothesis test should be conducted. Students should begin to see that logical arguments exist for structuring the hypotheses differently. In some interpretations of the problem, a two - tailed test can be appropriate for Par, Inc. In other interpretations of the same problem, a one - tailed test may be preferred. We suggest accepting different formulations of the Par, Inc. hypothesis test provided convincing rationale is provided.

Letting
1 = the population mean driving distance for the current golf ball
2 = the population mean driving distance for the new golf ball,

we suggest the following hypothesis test:

H0: 1 - 2 0

Ha: 1 - 2 > 0

This formulation is based on the information that the new golf ball is being designed to “resist cuts and yet still offer good driving distances.” The research hypothesis is not to prove the new golf ball out distances the current golf ball. In fact, Par could claim an improved quality with the cut resistance improvement provided the new golf ball has the same or better driving distance. The hypotheses have been structured so that rejection of H0 corresponds to the conclusion that the new golf ball has the lower mean driving distances; this conclusion indicates that the cut resistance advantage may be offset by the loss of distance.

...1. Think about a real world: Diagnosis Congestive heart failure (CHF), non compliance with daily weight or diet/situation.
Address using an independent or related samples t test.
2. Identify the independent (grouping) and dependant (response) variables important to study
3. Explain whether an independent sample or related sample t test is appropriate and why
4. Generate a hypothesis, including null and alternative hypothesis
5. Describe what information the effect size will tell you and what information the effect size will tell you and what information the p value or critical value approach will not
6. Using realistic numbers for the degrees of freedom, sample size and t statistic, report hypothetical results in 2-3 sentences
Solution: (1) Let’s consider the following research situation: The incidence of Congestive heart failure (CHF) is going to be studied based on two different diet groups: one group receives a special diet (a diet designed for preventing CHF), and a control group (which doesn’t receive any diet). We are interested in assessing whether there is a difference in the incidence of CHF for these two groups. In order to perform the analysis, a two-independent t-test will be used.
(2) In this case, the independent (grouping) variable is DIET, and the dependent (response) variable is CHF incidence rate.
(3) This analysis corresponds to an independent-samples design, because the treatments (diet/no diet) are applied to different...

...a population mean and a population proportion. Explain the meaning of an interval estimate of a population parameter.
An interval estimate for a specified population parameter (such as a mean or proportion) is a range of values in which the parameter is estimated to lie. In Chapter 6, you were assigned to find interval estimates for a population mean and a population proportion.
b) Is finding an interval estimate an example of inferential or descriptive statistics? Explain.
It is an interval estimate is an example of inferential statistics, as an estimate of the value of the population parameter is made based on sample statistics.
c) An interval estimate (23.8, 30.6) is determined for the mean age of NSCC students. Identify the point estimate and the margin of error of the interval estimate.
The point estimate=27.2 and the margin of error=3.4
d) The proportion of registered voters in Washington State in favor of a referendum to lower college tuitions is estimated with a 95% confidence level to be 45% with a margin of error ±3%. Is it possible the referendum will pass? Explain.
Yes, It is possible the referendum will pass. The margin of error is only stated with 95% confidence. In the given case, the estimate that the population proportion will lie between 0.42 and 0.48 (0.45 ± .03) is only stated with 95% certainty, so there is a...

...Chapter 10
StatisticalInferences Based on Two Samples
True/False
1. An independent sample experiment is an experiment in which there is no relationship between the measurements in the different samples.
Answer: True Difficulty: Medium
2. When testing the difference between two proportions selected from populations with large independent samples, the Z test statistic is used.
Answer: True Difficulty: Medium
3. In forming a large sample confidence interval for [pic], two assumptions are required: independent samples and sample sizes of at least 30.
Answer: True Difficulty: Medium
4. In testing the equality of population variances, two assumptions are required: independent samples and normally distributed populations.
Answer: True Difficulty: Medium
5. In an experiment involving matched pairs, a sample of 12 pairs of observations is collected. The degree of freedom for the t statistic is 10.
Answer: False Difficulty: Medium
6. When comparing two independent population means, if n1 = 13 and n2 = 10, degrees of freedom for the t statistic is 22.
Answer: False Difficulty: Easy
7. When comparing the variances of two normally distributed populations using independent random samples, if [pic], the calculated value of F will always be equal to one.
Answer: True Difficulty: Easy
8. In testing the difference between two population variances, it is a common practice to...

...a histogram of the 1990 returns.
(ii) Produce a histogram of the 1998 returns.
(iii) Find the mean, median, range and standard deviation for the 1990 returns.
Annual Returns % (1990)
Mean 12.91865979
Median 11.38
Standard Deviation 9.297513067
Range 75.01
(iv) Repeat part (iii) for the 1998 returns.
Annual Returns % (1998)
Mean 6.355463918
Median 5.4
Standard Deviation 5.170830853
Range 42.76
(v) Which was the better year for investors?
• 1990 was the better year for investors in regards to annual returns being consistent with the mean of 12.9% compare to 6.4% for 1998.
• The measure of variability was high in 1990 with the range of 75.01 compare to 42.76 for 1998. Another high variability for 1990 was the standard deviation of 9.30 compare to 5.17 for 1998.
(For Excel instructions see pages 28 and 61 of the textbook.)
Question 2. (StatisticalInferences: Single Population)
Feasibility Study: Companies that sell groceries over the Internet are called e-grocers. Customers enter their orders, pay by credit card and receive delivery by truck. To determine whether an e-grocery would be profitable in one large city, a potential e-grocer offered the service and recorded the size of the order for a random sample of customers. The data are stored in the data file.
(i) Estimate with 95% confidence the average order in the city.
Orders ($)
Mean...

...
1. A statistics professor has just given a final examination in his statisticalinference course. He is particularly interested in learning how his class of 40 students performed on this exam. The scores are shown below.
77 81 74 77 79 73 80 85 86 73
83 84 81 73 75 91 76 77 95 76
90 85 92 84 81 64 75 90 78 78
82 78 86 86 82 70 76 78 72 93
What is the mean score on this exam?
ANSWER:
Mean = 78.40, Median = 77.50
2. In February 2002 the Argentine peso lost 70% of its value compared to the United States dollar. This devaluation drastically raised the price of imported products. According to a survey conducted by AC Nielsen in April 2002, 68% of the consumers in Argentina were buying fewer products than before the devaluation, 24% were buying the same number of products, and 8% were buying more products. Furthermore, in a trend toward purchasing less-expensive brands, 88% indicated that they had changed the brands they purchased. Suppose the following complete set of results were reported. Use the following data to answer this question.
| |Number of Products Purchased | ...

...Decision rules, either sketch or write out.
Step 4
Conclusion
Mean of difference scores
T-statistic takes the same basic form (statistic minus expected value/SD)
Reported as t(9) = .85, n.s.
Statistical decision (don’t reject null all hypothesis are plausible; reject accept all alternative hypotheses)
Interpretation
Independent group t-tests
The logic of testing hypothesis about the means of two independent groups is the same as for previous statistical tests
Some minor calculation differences that can seem difficult at first
The test provides a more detailed discussion of the standard deviation
The equation for any test may be thought of as three parts
Sample statistic
Expected value (if H0 is true)
A measure of the variability in the sample statistic
H0 is written as the difference between two means
Two assumptions greatly simplify equations
Homogeneity of Variance: it is assumed that variance in population 1 Is equal to the variance in population 2.
IMPORTANT!!!
The assumption regards the population variances, not sample variances. It is possible that s21 is not equal to s22
Second assumption… Normality
CI for a single mean
For a one sample t-test
CI = M +/- (t-critical) (sm)
Critical value was a function of df and desired level of confidence
The logic of a CI for the difference between two means is identical...

...Inferences Concerning Two Means
If researchers want to compare two samples in terms of the mean scores using inferential statistics, they can utilize a confidence interval approach to the data or an approach that involves setting up and testing a null hypothesis. Whether two samples are considered to be independent or correlated is tied to the issue of the nature of the groups before data are collected on the study’s dependent variable. If the two groups have been assembled in such a way that a logical relationship exists between each member of the first sample and one and only one member of the second sample, then the two samples are correlated samples. However, if no such relationship exists, the two samples are independent samples.
Correlated samples come into existence in one of three ways. If a single group of people is measured twice (e.g., to provide pretest and posttest data), then a relationship exists in the data because each of the pretest scores goes with one and only one of the posttest scores, because both come from measuring the same research participant. A second situation that produces correlated samples is matching. Here, each person in the second group is recruited for the study because he or she is a good match for a particular individual in the first group. The matching could be done in terms of height, IQ, running speed, or any of a multitude of possible matching variables.
The matching variable,...

...The t-test is a hypothesis test. There are seven steps for a hypothesis test.
1.
2.
3.
4.
5.
6.
7.
State the null hypothesis
State the alternative hypothesis
State the level of significance
State the test statistic
Calculate
Statistical Conclusion
Experimental Conclusion
Example One-Sample t-test
The age of the head of household residents of Phoenix, Arizona has increased in the
recent years. Ten years ago the average age of head of household in Phoenix, Arizona
was 33. A random sample of 40 head of household residents in Phoenix, Arizona was
taken and the ages were recorded as follows.
25
46
63
37
72
20
47
41
59
64
32
37
80
59
63
31
38
35
45
36
45
22
36
44
48
69
31
36
34
37
70
33
65
29
31
76
51
25
27
21
27
We wish to use this data to test if the hypothesis that the average age of head of
household residents in Phoenix, Arizona has increased.
This is a hypothesis test so we will need to go through the seven steps of a hypothesis
testing.
Step 1: Null Hypothesis
Since in the past the mean Head of Household age was 33 that is the statement of no
effect which is what the null hypothesis gives.
H 0 : 33
Step 2: Alternative Hypothesis
We want to know if the mean head of household age has increased so the alternative is
H A : 33
This is a one-tailed test as we are only looking at a one-sided alternative.
Step 3: Level of Significance
...

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