Question

Math

Posted 6 months ago

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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. $\mathrm{H}_{\mathrm{a}}: \beta \neq 0$ 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.
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Answer from Sia

Posted 6 months ago

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.

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