Question

1. a study is being done on the average number of texts sent by a high school student per hour. a random sample of size 30 produced a sample mean of 19.7 texts per hour. a simulation was conducted assuming mean number of texts sent per hour is 19.7. 200 random samples of size 30 were selected, and the results are displayed below.

a. based on this simulation, find the margin of error for a 95% interval to estimate the mean number of texts sent per hour.

b. calculate a 95% interval that could be used to estimate the mean number of texts sent per hour.

c. national statistics claim that average amount of texts sent per hour is 16.5. is this result consistent with the interval you calculated in part b? explain.

Explanation:

Step1: Recall margin - of - error formula for 95% confidence interval

For a 95% confidence interval from a sampling distribution, the margin of error $E$ is approximately $2\times$ the standard deviation of the sampling distribution. Given that the standard deviation of the sampling distribution (SD) is $2.121$, the formula is $E = 2\times SD$.
$E=2\times2.121$

Step2: Calculate the margin of error

$E = 4.242$

Step3: Calculate the 95% confidence interval

The formula for a confidence interval is $\bar{x}\pm E$, where $\bar{x}$ is the sample mean. Here, $\bar{x}=19.7$. The lower limit is $19.7 - 4.242=15.458$ and the upper limit is $19.7 + 4.242 = 23.942$. So the 95% confidence interval is $(15.458,23.942)$.

Step4: Check consistency

The national statistic claim is that the average number of texts sent per hour is $16.5$. Since $16.5$ lies within the interval $(15.458,23.942)$, the result is consistent with the interval calculated in part b.

Answer:

a. $4.242$
b. $(15.458,23.942)$
c. Yes, because $16.5$ lies within the interval $(15.458,23.942)$.