to test the effect of a physical fitness course on ones physical ability, the number of sit-ups that a person could do in one minute, both before and after the course, was recorded. ten individuals are randomly selected to participate in the course. the results are displayed in the following table. using this data, find the 90% confidence interval for the true difference in the number of sit-ups each person can do before and after the course. assume that the numbers of sit-ups are normally distributed for the population both before and after completing the course.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
sit-ups before & 34 & 44 & 20 & 33 & 23 & 49 & 31 & 50 & 39 & 34 \\
\hline
sit-ups after & 51 & 50 & 25 & 47 & 38 & 53 & 44 & 56 & 43 & 48 \\
\hline
\end{tabular}

step 4 of 4: construct the 90% confidence interval. round your answers to one decimal place.

Question

to test the effect of a physical fitness course on ones physical ability, the number of sit-ups that a person could do in one minute, both before and after the course, was recorded. ten individuals are randomly selected to participate in the course. the results are displayed in the following table. using this data, find the 90% confidence interval for the true difference in the number of sit-ups each person can do before and after the course. assume that the numbers of sit-ups are normally distributed for the population both before and after completing the course.

\

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \\hline sit-ups before & 34 & 44 & 20 & 33 & 23 & 49 & 31 & 50 & 39 & 34 \\\\ \\hline sit-ups after & 51 & 50 & 25 & 47 & 38 & 53 & 44 & 56 & 43 & 48 \\\\ \\hline \\end{tabular}

step 4 of 4: construct the 90% confidence interval. round your answers to one decimal place.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
sit-ups before & 34 & 44 & 20 & 33 & 23 & 49 & 31 & 50 & 39 & 34 \\
\hline
sit-ups after & 51 & 50 & 25 & 47 & 38 & 53 & 44 & 56 & 43 & 48 \\
\hline
\end{tabular}

step 4 of 4: construct the 90% confidence interval. round your answers to one decimal place.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Paired t-Interval", "Mean Difference", "Margin of Error" ], "new_concepts": [], "current_concepts": [ "Paired t-Interval", "Mean Difference", "Margin of Error" ] } ## Calculate paired differences $$ \begin{aligned} d &= \text{Before} - \text{After} \ d_1 &= 34 - 51 = -17 \ d_2 &= 44 - 50 = -6 \ d_3 &= 20 - 25 = -5 \ d_4 &= 33 - 47 = -14 \ d_5 &= 23 - 38 = -15 \ d_6 &= 49 - 53 = -4 \ d_7 &= 31 - 44 = -13 \ d_8 &= 50 - 56 = -6 \ d_9 &= 39 - 43 = -4 \ d_{10} &= 34 - 48 = -14 \end{aligned} $$ ## Compute sample mean and standard deviation of differences $$ \begin{aligned} n &= 10 \ \sum d_i &= -98 \ \bar{d} &= \frac{-98}{10} = -9.8 \ \sum d_i^2 &= (-17)^2 + (-6)^2 + (-5)^2 + (-14)^2 + (-15)^2 + (-4)^2 + (-13)^2 + (-6)^2 + (-4)^2 + (-14)^2 \ &= 289 + 36 + 25 + 196 + 225 + 16 + 169 + 36 + 16 + 196 = 1204 \ s_d &= \sqrt{\frac{\sum d_i^2 - n\bar{d}^2}{n-1}} = \sqrt{\frac{1204 - 10(-9.8)^2}{9}} = \sqrt{\frac{1204 - 960.4}{9}} = \sqrt{\frac{243.6}{9}} \approx 5.20256 \end{aligned} $$ ## Calculate confidence interval endpoints $$ \begin{aligned} df &= 9,\quad t_{0.05, 9} = 1.833 \ E &= t^* \cdot \frac{s_d}{\sqrt{n}} = 1.833 \cdot \frac{5.20256}{\sqrt{10}} \approx 1.833 \cdot 1.64519 \approx 3.0156 \ \text{Lower Endpoint} &= \bar{d} - E = -9.8 - 3.0156 \approx -12.8 \ \text{Upper Endpoint} &= \bar{d} + E = -9.8 + 3.0156 \approx -6.8 \end{aligned} $$ Lower endpoint: -12.8 Upper endpoint: -6.8 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Paired t-Interval" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate paired differences

Compute sample mean and standard deviation of differences

Calculate confidence interval endpoints

Answer: