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.RE.18c

Capitolo 8, Problema 8.RE.18c

In Exercises 17 and 18, (c) find the standardized test statistic t, Assume the samples are random and independent, and the populations are normally distributed.

A real estate agent claims that there is no difference between the mean household incomes of two neighborhoods. The mean income of 12 randomly selected households from the first neighborhood is \$52,750 with a standard deviation of \$2900. In the second neighborhood, 10 randomly selected households have a mean income of \$51,200 with a standard deviation of \$2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal.

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Identify the hypotheses for the test: the null hypothesis $H_0$ states that the mean incomes are equal, i.e., $\mu_1 = \mu_2$, and the alternative hypothesis $H_a$ states that the means are different, i.e., $\mu_1 \neq \mu_2$.

Since the population variances are assumed equal, calculate the pooled standard deviation $s_p$ using the formula: \[s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\] where $n_1$ and $n_2$ are the sample sizes, and $s_1$ and $s_2$ are the sample standard deviations.

Calculate the standard error of the difference between the two sample means using the pooled standard deviation: \[SE = s_p \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\]

Compute the test statistic $t$ using the formula: \[t = \frac{\bar{x}_1 - \bar{x}_2}{SE}\] where $\bar{x}_1$ and $\bar{x}_2$ are the sample means.

Determine the degrees of freedom for the test as $df = n_1 + n_2 - 2$, then compare the calculated $t$ value to the critical $t$ value from the $t$-distribution table at $\alpha = 0.01$ for a two-tailed test to decide whether to reject the null hypothesis.

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Two-Sample t-Test for Means

This test compares the means of two independent samples to determine if there is a statistically significant difference between their population means. It is appropriate when samples are random, independent, and populations are normally distributed. The test statistic follows a t-distribution under the null hypothesis.

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Sampling Distribution of Sample Mean

Pooled Variance and Equal Variance Assumption

When population variances are assumed equal, the sample variances are combined into a pooled variance estimate. This pooled variance provides a more accurate estimate of the common variance, which is used to calculate the standard error of the difference between means in the t-test.

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Variance & Standard Deviation of Discrete Random Variables

Significance Level (α) and Hypothesis Testing

The significance level α (here 0.01) defines the threshold for rejecting the null hypothesis. If the calculated t-statistic falls in the critical region beyond the t-distribution cutoff, the null hypothesis (no difference in means) is rejected, indicating a statistically significant difference.

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Performing Hypothesis Tests: Proportions

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