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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
Tutti i libri di testoLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblema 8.RE.14
Capitolo 8, Problema 8.RE.14

In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1>= μ2; α=0.01. Assume (σ1)^2 = (σ2)^2


Sample statistics: x̅1= 44.5, s1= 5.85, n1= 17 and x̅2= 49.1, s2= 5.25, n2= 18

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Identify the null and alternative hypotheses based on the claim. Since the claim is \( \mu_1 \geq \mu_2 \), the hypotheses are: \( H_0: \mu_1 - \mu_2 \geq 0 \) and \( H_a: \mu_1 - \mu_2 < 0 \). This is a left-tailed test.
Since the population variances are assumed equal (\( \sigma_1^2 = \sigma_2^2 \)), calculate the pooled sample variance using the formula: \[ S_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \]
Calculate the test statistic \( t \) using the pooled variance: \[ t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \] Note that the hypothesized difference under \( H_0 \) is 0.
Determine the degrees of freedom for the test, which is \( df = n_1 + n_2 - 2 \). Then, find the critical value \( t_{\alpha} \) from the t-distribution table corresponding to \( \alpha = 0.01 \) and the calculated degrees of freedom for a left-tailed test.
Compare the calculated test statistic \( t \) to the critical value. If \( t \) is less than the critical value, reject the null hypothesis; otherwise, do not reject it. This will help you decide whether there is sufficient evidence to support the claim \( \mu_1 \geq \mu_2 \).

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Hypothesis Testing for Two Population Means

This involves formulating null and alternative hypotheses about the difference between two population means (μ1 and μ2). The goal is to use sample data to decide whether to reject the null hypothesis at a given significance level (α), based on evidence from the test statistic.
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Percorso guidato
08:24
Difference in Means: Hypothesis Tests

Assumption of Equal Population Variances

When the population variances are assumed equal (σ1² = σ2²), a pooled variance estimate is used to calculate the test statistic. This assumption simplifies the test and affects the degrees of freedom, leading to a more precise inference about the difference between means.
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Percorso guidato
04:48
Population Standard Deviation Known

Significance Level and Decision Rule

The significance level (α = 0.01) defines the probability of rejecting the null hypothesis when it is true (Type I error). It sets the critical value(s) for the test statistic, guiding whether the observed data provide enough evidence to support the claim that μ1 ≥ μ2.
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Percorso guidato
03:53
Conditional Probability Rule
Pratica correlata
Domanda del libro di testo

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.


Sample 1: The fuel efficiencies of 12 cars

Sample 2: The fuel efficiencies of the same 12 cars using an alternative fuel




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Domanda del libro di testo

In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1≠μ2; α=0.05


Population statistics: σ1= 14 and σ2= 15


Sample statistics: x̅1 = 87, n1 = 410, and x̅2= 85, n2= 340

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Domanda del libro di testo

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.


Sample 1: The weights of 45 oranges

Sample 2: The weights of 40 grapefruits


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Domanda del libro di testo

In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1≠ μ2; α=0.01. Assume (σ1)^2 = (σ2)^2


Sample statistics: x̅1= 61, s1= 3.3, n1= 5 and x̅2= 55.1, s2= 1.2, n2= 7

50
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Domanda del libro di testo

In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1< μ2; α=0.10. Assume (σ1)^2 ≠ (σ2)^2


Sample statistics: x̅1=0.015, s1=0.011, n1= 8 and x̅2=0.019, s2=0.004, n2= 6

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Domanda del libro di testo

In Exercises 17 and 18, (a) identify the claim and state Ho and Ha, 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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