17. In a modeling agency, a researcher wishes to see if the average height of female models is less than 67 inches, as the coach claims. A random sample of 60 models has an average height of 65.8 inches. The standard deviation of the sample is 1.7 inches. At 𝛼 = 0.05, which of the following shows the appropriate rejection of the given problem?*A.B.C.D.
Question
17. In a modeling agency, a researcher wishes to see if the average height of female models is less than 67 inches, as the coach claims. A random sample of 60 models has an average height of 65.8 inches. The standard deviation of the sample is 1.7 inches. At 𝛼 = 0.05, which of the following shows the appropriate rejection of the given problem?*A.B.C.D.
Solution
To answer this question, we need to conduct a one-sample t-test. Here are the steps:
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State the null hypothesis (H0) and the alternative hypothesis (H1). In this case, H0: μ = 67 (the average height of female models is 67 inches) and H1: μ < 67 (the average height of female models is less than 67 inches).
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Calculate the test statistic. The formula for the test statistic in a one-sample t-test is (x̄ - μ) / (s/√n), where x̄ is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n is the sample size. Plugging in the given values, we get (65.8 - 67) / (1.7/√60) = -5.26.
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Determine the critical value for α = 0.05 in a one-tailed t-test with 59 degrees of freedom (n - 1 = 60 - 1 = 59). Using a t-table or a t-distribution calculator, we find that the critical value is approximately -1.671.
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Compare the test statistic to the critical value. If the test statistic is less than the critical value, we reject the null hypothesis. In this case, -5.26 is less than -1.671, so we reject the null hypothesis.
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Therefore, the appropriate rejection for the given problem is that the average height of female models is less than 67 inches.
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