Chapter 10.3, Problem 13E

(a)

To determine

To construct a 95% confidence, interval for the mean.

Answer to Problem 13E

The $95\%$ confidence interval for the mean increase in height for boys between the ages of $12\text{ and 14}$ is $\left(5.182\text{ inches, 5}\text{.5529 inches}\right)$

Explanation of Solution

Given information:

Age 12	Age 14	Age 12	Age 14
57.9	62.9	55.4	61.8
61.1	65.9	58.7	64.2
62.7	67.5	64.3	69.8
67.5	73.7	58.1	63.3
59.2	64.9	63.3	69.6
61.4	67	61.2	66.4
60.7	65.5	64.5	69.2
55.9	62.1	55.9	62
59.7	65.4	60.4	65.7
56.3	61.5	57.8	62.8
63	68.5	68.3	73.6
58.6	63.9	63	67.7
61.1	65.8	64.4	69.2
59.5	64.5	58.2	64.6
61.6	66.3	59.7	66.1
59.3	64.2	60.2	65.9
62.1	67.1	63.7	68.3
62.8	68.1	60.2	65.2
65.3	69.9	62.7	67.9
60.4	66.7	55.6	61.73

Formula used Construct the $95\%$ confidence interval for the mean increase in height for boys between the ages of $12\text{ and 14}$ by using MINITAB.

MINITAB procedure:

Step $1:$ Choose Stat $>$ Basic Statistics $>$ Paired $t$

Step $2:$ In First Sample enter the column of Age $14$

Step $3:$ In Second Sample enter the column of Age $12$.

Step $4:$ Check Options, and enter the Confidence level as $95.0$

Step $5:$ Choose not equal in alternative

Step $6:$ Click OK in all dialog boxes.

Calculation:

MINITAB output:

Paired T-Test and $\text{CI}:\text{Age 14, Age 12}$

Paired $\text{T for Age 14}-\text{Age}12$.

$\begin{array}{l}\text{N}\text{Mean StDev SE Mean}\\ \text{Age 14}\text{40}66.160\text{ 2}\text{.949 0}\text{.466}\\ \text{Age 12}\text{ 40}\text{ 60}\text{.792 }\text{ 3}\text{.097 0}\text{.490}\\ \text{Difference 40 5}\text{.3675 0}\text{.5797 0}\text{.0917 }\end{array}$

$\text{95\% CI}$ for mean difference: $\left(5.1821,\text{ 5}\text{.5529}\right)$

T-test of mean difference $=0\left(\text{vs not }=0\right):\text{ T-value}=58.56,\text{ P-value}=0.000$

Form the above MINITAB output, the $95\%$ confidence interval for the mean increase in height for boys between the ages of $12\text{ and 14}$ is $\left(5.182\text{ inches, 5}\text{.5529 inches}\right)$.

(b)

To determine

To explain whether the confidence interval contradict the claim that mean height is 5.5 inches.

Answer to Problem 13E

No. tile confidence interval does not contradicts the claims that “the mean increase in height is $5.5$ inches” because the confidence interval contain $5.5$ inches.

Explanation of Solution

Calculation:

Check whether the confidence interval contradict the given claim or not.

The $95\%$ t-confidence interval for $\left({\mu }_1-{\mu }_2\right)\text{ is }\left(5.1821,5.5529\right)$ which contain the hypothesized value $5.5$ inches. That is, the hypothesized value $5.5$ inches lies inside the interval $\left(5.1821,5.5529\right)$

No. the confidence interval does not contradicts the claims that “the mean increase in height is $5.5$ Inches” because the confidence interval contain $5.5$ inches.