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7.5 Hypothesis Testing for Variance and Standard Deviation

7.5 Hypothesis Testing for Variance and Standard Deviation. Find critical values for a χ² -test Use the χ² -test to test a variance or a standard deviation. Find the critical χ² -value for a right-tailed test when n = 18 and α = 0.01. d.f . = 17. Critical value: 33.409.

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7.5 Hypothesis Testing for Variance and Standard Deviation

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  1. 7.5 Hypothesis Testing for Variance and Standard Deviation

    Find critical values for a χ²-test Use the χ²-test to test a variance or a standard deviation
  2. Find the critical χ²-value for a right-tailed test when n = 18 and α = 0.01. d.f. = 17 Critical value: 33.409 Try it yourself 1 Finding Critical Values for χ²
  3. Find the critical χ²-value for a left-tailed test when n = 30 and α = 0.05. d.f. = 29 Critical value: 17.708 Try it yourself 2 Finding Critical Values for χ²
  4. Find the critical χ²-value for a two-tailed test when n = 51 and α = 0.01. d.f. = 50 Critical values: 27.991 and 79.490 Try it yourself 3 Finding Critical Values for χ²
  5. The χ²-test for a variance or standard deviation is a statistical test for a population variance or standard deviation. The χ²-test can be used when the population is normal. The test statistic is s² and the standardized test statistic follows a chi-square distribution with degrees of freedom d.f. = n -1 χ²-Test for a Variance σ² or Standard Deviation σ
  6. A bottling company claims that the variance of the amount of sports drink in a 12-ounce bottle is no more than 0.40. A random sample of 31 bottles has a variance of 0.75. At α = 0.01, is there enough evidence to reject the company’s claim? Assume the population is normally distributed. Critical value: 50.892; Rejection region: χ² > 50.892 χ² = 56.250 Reject the null hypothesis There is enough evidence at the 1% level of significance to reject the claim that the variance of the amount of sports drink in a 12-ounce bottle is no more than 0.40. Try it yourself 4 Using a Hypothesis Test for the Population Variance
  7. A police chief claims that the standard deviation of the lengths of response times is less than 3.7 minutes. A random sample of 9 response times has a standard deviation of 3.0 minutes. At α = 0.05, is there enough evidence to support the police chief’s claim? Assume the population is normally distributed. Critical value: 2.733; Rejection region: χ² < 2.733 χ² = 5.259 Fail to reject the null hypothesis. There is not enough evidence at the 5% level of significance to support the claim that the standard deviation of lengths of response times is less than 3.7 minutes. Try it yourself 5 Using a Hypothesis Test for the Standard Deviation
  8. A company that offers dieting products and weight loss services claims that the variance of the weight losses of their users is 25.5. A random sample of 13 users has a variance of 10.8. At α = 0.10, is there enough evidence to reject the company’s claim? Assume the population is normally distributed. Critical values: 5.226 and 21.026 Rejection region: χ² < 5.226, χ² > 21.026 χ² = 5.082 Reject the null hypothesis. There is enough evidence at the 10% level of significance to reject the claim that the variance of the weight losses of users is 25.5. Try it yourself 6 Using a Hypothesis Test for the Population Variance
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