Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problema 8.2.1

Capitolo 8, Problema 8.2.1

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

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Ensure that the data for both populations are approximately normally distributed. This is particularly important if the sample sizes are small (n < 30). If the sample sizes are large, the Central Limit Theorem allows for some relaxation of this condition.

Verify that the samples are independent of each other. This means that the selection of one sample does not influence the selection of the other sample.

Check whether the population variances are equal or unequal. This will determine whether you use the pooled variance t-test (equal variances) or the Welch's t-test (unequal variances).

Confirm that the data is measured on an interval or ratio scale, as the t-test requires numerical data for meaningful comparison.

Ensure that the sample sizes are not too small to provide sufficient statistical power for detecting a difference, unless the effect size is expected to be large.

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Normality

For the t-test to be valid, the data from both populations should ideally follow a normal distribution. This is particularly important when sample sizes are small (typically less than 30), as the t-test relies on the assumption that the sampling distribution of the mean is approximately normal.

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Finding Z-Scores for Non-Standard Normal Variables

Independence

The observations in each sample must be independent of each other. This means that the selection of one observation should not influence the selection of another. Violating this assumption can lead to biased results and incorrect conclusions.

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Independence Test

Equal Variances

The t-test assumes that the variances of the two populations being compared are equal. This is known as the assumption of homogeneity of variance. If this assumption is violated, a modified version of the t-test, such as Welch's t-test, may be more appropriate.

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Variance & Standard Deviation of Discrete Random Variables

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What conditions are necessary in order to use the z-test to test the difference between two population means?

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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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[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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