Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

문제 8.3.1

8장, 문제 8.3.1

What conditions are necessary to use the dependent samples t-test for the mean of the differences for a population of paired data?

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The dependent samples t-test (also known as the paired samples t-test) is used when comparing the means of two related groups. The first condition is that the data must consist of paired observations, meaning each data point in one group is uniquely matched to a data point in the other group (e.g., pre-test and post-test scores for the same individuals).

The second condition is that the differences between the paired observations (calculated as the difference between each pair) should be approximately normally distributed. This can be checked using a normality test (e.g., Shapiro-Wilk test) or by examining a histogram or Q-Q plot of the differences.

The third condition is that the data should be measured on an interval or ratio scale, meaning the differences between values are meaningful and consistent (e.g., test scores, weights, or times).

The fourth condition is that the sample of paired differences should be randomly selected from the population of interest to ensure the results are generalizable.

Finally, the sample size should be sufficiently large to provide reliable results. If the sample size is small, the normality assumption becomes more critical, and you may need to rely on robust statistical methods or transformations if normality is violated.

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Dependent Samples

Dependent samples, also known as paired samples, refer to data sets where the observations are related or matched in some way. This often occurs in before-and-after studies or when measuring the same subjects under different conditions. Understanding this concept is crucial for applying the dependent samples t-test, as it ensures that the analysis accounts for the inherent relationship between the paired observations.

Normality of Differences

For the dependent samples t-test to be valid, the differences between the paired observations should be approximately normally distributed. This assumption is important because the t-test relies on the properties of the normal distribution to make inferences about the population mean. If the sample size is large, the Central Limit Theorem may allow for some flexibility regarding this assumption.

Homogeneity of Variance

While the dependent samples t-test does not require the same homogeneity of variance assumption as independent samples t-tests, it is still important to consider the variability of the differences. The test assumes that the variance of the differences is constant across the pairs. If there is significant variability, it may affect the reliability of the test results, necessitating further investigation or alternative statistical methods.

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What conditions are necessary to use the t-test for testing the difference between two population means?

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[APPLET] Teaching Methods

A new method of teaching reading is being tested on third grade students. A group of third grade students is taught using the new curriculum. A control group of third grade students is taught using the old curriculum. The reading test scores for the two groups are shown in the back-to-back stem-and-leaf plot.

At , α=0.10 is there enough evidence to support the claim that the new method of teaching reading produces higher reading test scores than the old method does? Assume the population variances are equal.

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Blue Crabs A marine researcher claims that the stomachs of blue crabs from one location contain more fish than the stomachs of blue crabs from another location. The stomach contents of a sample of 25 blue crabs from Location A contain a mean of 320 milligrams of fish and a standard deviation of 60 milligrams. The stomach contents of a sample of 15 blue crabs from Location B contain a mean of 280 milligrams of fish and a standard deviation of 80 milligrams. At , α= 0.01can you support the marine researcher’s claim? Assume the population variances are equal.

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Explain how to perform a two-sample z-test for the difference between two population means using independent samples with and known.

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"Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

[APPLET] Precipitation A climatologist claims that the precipitation in Seattle, Washington, was greater than in Birmingham, Alabama, in a recent year. The daily precipitation amounts (in inches) for 30 days in a recent year in Seattle are shown below. Assume the population standard deviation is 0.25 inch.

0.00 0.00 0.05 0.01 0.21 0.00 0.00 0.52 0.00 0.010.00 0.19 0.00 0.18 0.02 0.02 0.13 0.00 0.03 0.000.04 0.00 0.41 0.23 0.00 0.80 0.15 0.00 0.00 0.79

The daily precipitation amounts (in inches) for 30 days in a recent year in Birmingham are shown below. Assume the population standard deviation is 0.52 inch.

0.00 0.96 0.84 0.00 0.10 0.00 0.00 0.20 0.00 0.54 0.97 0.00 0.35 0.02 0.04 0.70 0.00 0.00 0.00 0.00 0.03 0.01 0.15 0.27 0.00 0.00 0.93 0.00 0.89 0.01

At α=0.05, can you support the climatologist’s claim? (Source: NOAA)"

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Annual Income

A politician claims that the mean household income in a recent year is greater in York County, South Carolina, than it is in Elmore County, Alabama. In York County, a sample of 23 residents has a mean household income of \$64,900 and a standard deviation of \$16,000. In Elmore County, a sample of 19 residents has a mean household income of \$59,500 and a standard deviation of \$23,600. At , α= 0.05can you support the politician’s claim? Assume the population variances are not equal. (Adapted from U.S. Census Bureau)

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