Domanda del libro di testo
In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p <0.12, α=0.01. Sample statistics: p_hat = 0.10, n=40
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Ch. 7 - Hypothesis Testing with One Sample
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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TOC.tableOfContents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Capitolo 7, Problema 7.5.19
In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.
Claim: σ<40, α=0.01 . Sample statistics: s=40.8, n=12
Guida verificata passo dopo passo1
Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: σ ≥ 40, and the alternative hypothesis is H₁: σ < 40. This is a left-tailed test since the claim is that the population standard deviation is less than 40.
Step 2: Determine the test statistic to use. Since this is a test about the population standard deviation, use the chi-square test statistic formula: χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s is the sample standard deviation, and σ₀ is the hypothesized population standard deviation.
Step 3: Substitute the given values into the formula. Here, n = 12, s = 40.8, and σ₀ = 40. Calculate s² (the sample variance) as s² = (40.8)², and then compute χ² using the formula.
Step 4: Determine the critical value for the chi-square distribution. Use the chi-square table with degrees of freedom (df) = n - 1 = 12 - 1 = 11 and a significance level of α = 0.01 for a left-tailed test. Find the critical value χ²_critical corresponding to these parameters.
Step 5: Compare the calculated χ² value to the critical value χ²_critical. If χ² < χ²_critical, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim about the population standard deviation.
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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), then using sample statistics to determine whether to reject H0 in favor of H1. In this case, the null hypothesis would state that the population standard deviation is greater than or equal to 40, while the alternative claims it is less than 40.
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Step 1: Write Hypotheses
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 0.01, indicating a 1% risk of concluding that the population standard deviation is less than 40 when it is not. This level of significance helps determine the threshold for making statistical inferences based on the sample data.
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03:33Finding Binomial Probabilities Using TI-84 Example 1
Chi-Square Test for Variance
The Chi-Square test for variance is a statistical test used to determine if the variance of a population is equal to a specified value. It is particularly useful when the population is normally distributed. In this case, the test will compare the sample variance (derived from the sample standard deviation) to the hypothesized population variance to assess whether the claim about the population standard deviation being less than 40 holds true.
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Domanda del libro di testo
Hypothesis Testing Using Rejection Regions In Exercises 23–30, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic X^2, (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 population is normally distributed.
Salaries The annual salaries (in dollars) of 15 randomly chosen senior level graphic design specialists are shown in the table at the left. At α=0.05, is there enough evidence to support the claim that the standard deviation of the annual salaries is different from \$13,056?
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Domanda del libro di testo
In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.
Claim: ; μ ≤ 22,500; α = 0.01; α = 1200
Sample statistics: x_bar = 23,500, n = 45
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Domanda del libro di testo
In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.
Claim: ; μ ≠ 5880; α = 0.03; α = 413
Sample statistics: x_bar = 5771, n = 67
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Domanda del libro di testo
Explain how to test a population variance or a population standard deviation.
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Domanda del libro di testo
In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.
Left-tailed test, n=24,α=0.05
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