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Ch. 7 - Hypothesis Testing with One Sample
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. 7 - Hypothesis Testing with One SampleProblema 7.RE.55
Capitolo 7, Problema 7.RE.55

In Exercises 55–58, test the claim about the population variance or standard deviation at the level of significance . Assume the population is normally distributed.


Claim: σ^2 > 2; α=0.10. Sample statistics: s^2 = 2.95, n=18

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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: σ² ≤ 2, and the alternative hypothesis is H₁: σ² > 2. This is a right-tailed test since the claim is that the population variance is greater than 2.
Step 2: Determine the test statistic formula for a chi-square test for variance. The formula is χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s² is the sample variance, and σ₀² is the hypothesized population variance.
Step 3: Substitute the given values into the formula. Here, n = 18, s² = 2.95, and σ₀² = 2. Compute χ² = ((18 - 1) * 2.95) / 2.
Step 4: Determine the critical value for the chi-square distribution at α = 0.10 with degrees of freedom df = n - 1 = 17. Use a chi-square distribution table or statistical software to find the critical value for a right-tailed test.
Step 5: Compare the calculated χ² value to the critical value. If χ² > critical value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Conclude whether there is sufficient evidence to support the claim that the population variance is greater than 2.

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Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1). In this case, the null hypothesis would state that the population variance is less than or equal to 2, while the alternative hypothesis claims that it is greater than 2. The outcome of the test determines whether to reject or fail to reject the null hypothesis.
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Significance Level (α)

The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.10, meaning there is a 10% risk of concluding that the population variance is greater than 2 when it is not. This threshold helps determine the critical value for the test statistic, guiding the decision-making process.
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Chi-Square Distribution

The Chi-Square distribution is a statistical distribution commonly used in hypothesis testing for variance and standard deviation. It is particularly relevant when the population is normally distributed, as is the case here. The test statistic for variance is calculated using the sample variance and the sample size, and it follows a Chi-Square distribution with degrees of freedom equal to n-1. This distribution helps determine the critical value needed to assess the hypothesis.
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