Nancy obtained a random sample of students and noticed a positive linear relationship between their ages and the amount of water (in $\mathrm{mL}$ ) they said they drank yesterday. A $95 \%$ confidence interval for the slope of the regression line was $(16.3,154.8)$.

Nancy wants to use this interval to test $\mathrm{H}_{0}: \beta=0$ vs. $\mathrm{H}_{\mathrm{a}}: \beta
eq 0$ at the $\alpha=0.05$ level of significance. Assume that all conditions for inference have been met.

Which of these is the most appropriate conclusion for this population of students?

Choose 1 answer:
(A) Reject $\mathrm{H}_{0}$. Nancy can't conclude a linear relationship between age and water consumption.
(B) Fail to reject $\mathrm{H}_{0}$. Nancy can't conclude a linear relationship between age and water consumption.
(C) Reject $\mathrm{H}_{0}$. This suggests a linear relationship between age and water consumption.
(D) Fail to reject $\mathrm{H}_{0}$. This suggests a linear relationship between age and water consumption.

Question

Nancy obtained a random sample of students and noticed a positive linear relationship between their ages and the amount of water (in $\mathrm{mL}$ ) they said they drank yesterday. A $95 \%$ confidence interval for the slope of the regression line was $(16.3,154.8)$.

Nancy wants to use this interval to test $\mathrm{H}_{0}: \beta=0$ vs. $\mathrm{H}_{\mathrm{a}}: \beta 
eq 0$ at the $\alpha=0.05$ level of significance. Assume that all conditions for inference have been met.

Which of these is the most appropriate conclusion for this population of students?

Choose 1 answer:
(A) Reject $\mathrm{H}_{0}$. Nancy can't conclude a linear relationship between age and water consumption.
(B) Fail to reject $\mathrm{H}_{0}$. Nancy can't conclude a linear relationship between age and water consumption.
(C) Reject $\mathrm{H}_{0}$. This suggests a linear relationship between age and water consumption.
(D) Fail to reject $\mathrm{H}_{0}$. This suggests a linear relationship between age and water consumption.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the null hypothesis and the alternative hypothesis for the hypothesis test. ‖ ‖ step 2 ⋮ Examine the 95% confidence interval for the slope of the regression line. ‖ ‖ step 3 ⋮ Determine if the null hypothesis value (slope = 0) falls within the confidence interval. ‖ ‖ step 4 ⋮ Make a conclusion based on whether the null hypothesis value is inside or outside the confidence interval. ‖ ∻Answer∻ ⚹ (C) Reject $ \mathrm{H}_{0} $. This suggests a linear relationship between age and water consumption. ⚹ ∻Key Concept∻ ⚹Confidence Interval and Hypothesis Testing⚹ ∻Explanation∻ ⚹If the confidence interval for the slope does not contain the null hypothesis value (in this case, slope = 0), we reject the null hypothesis. Since the interval (16.3, 154.8) does not include 0, we reject $ \mathrm{H}_{0} $ and conclude there is a linear relationship.⚹◊What is the formula to calculate the test statistic for the hypothesis test in this scenario?⍭ Generate me a similar question◊

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