Question

the following data show the number of hours per day 12 adults spent in front of screens watching television - related content. complete parts a and b below. 1.7 4.7 3.7 5.2 7.6 7.3 5.3 2.2 5.1 1.7 2.3 8.5 a. construct a 99% confidence interval to estimate the average number of hours per day adults spend in front of screens watching television - related content. the 99% confidence interval to estimate the average number of hours per day adults spend in front of screens watching television - related content is from hours to hours. (round to two decimal places as needed.)

Explanation:

Step1: Calculate sample mean

First, sum all the data values: $1.7 + 4.7+3.7 + 5.2+7.6+7.3+5.3+2.2+5.1+1.7+2.3+8.5 = 55.3$. Then, divide by the sample size $n = 12$. The sample mean $\bar{x}=\frac{55.3}{12}\approx4.61$.

Step2: Calculate sample standard - deviation

Use the formula $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
First, calculate $(x_{i}-\bar{x})^{2}$ for each data point. For example, for $x_1 = 1.7$, $(1.7 - 4.61)^{2}=(-2.91)^{2}=8.4681$. After calculating for all 12 data points and summing them up, we get $\sum_{i = 1}^{12}(x_{i}-\bar{x})^{2}\approx57.97$. Then $s=\sqrt{\frac{57.97}{12 - 1}}\approx2.29$.

Step3: Determine the critical value

Since the sample size $n = 12$ (small - sample, $n<30$) and we want a 99% confidence interval, the degrees of freedom $df=n - 1=11$. Looking up in the t - distribution table, the critical value $t_{\alpha/2}$ for a 99% confidence interval with $df = 11$ is $t_{0.005,11}=3.106$.

Step4: Calculate the margin of error

The formula for the margin of error $E=t_{\alpha/2}\frac{s}{\sqrt{n}}$. Substitute $t_{\alpha/2}=3.106$, $s = 2.29$, and $n = 12$ into the formula: $E=3.106\times\frac{2.29}{\sqrt{12}}\approx2.04$.

Step5: Calculate the confidence interval

The confidence interval is $\bar{x}-E<\mu<\bar{x} + E$. Substitute $\bar{x}=4.61$ and $E = 2.04$: $4.61-2.04<\mu<4.61 + 2.04$, which gives $2.57<\mu<6.65$.

Answer:

The 99% confidence interval to estimate the average number of hours per day adults spend in front of screens watching television - related content is from $2.57$ hours to $6.65$ hours.