Question

a survey about study habits is administered to a random sample of 50 college students. one question asks how, on average, the students study per day. the results were found to be normally - distributed. the mean of the responses to this question is 4.2 hours and the standard deviation 1.7 hours. which range gives the 95% confidence interval for the true population mean if the standard error is 0.24? a. 3.72 to 4.67 hours b. 2.32 to 5.61 hours c. 2.50 to 5.90 hours d. 3.96 to 4.44 hours

Explanation:

Step1: Recall confidence - interval formula for large samples

For a 95% confidence interval of the population mean when the sample size $n\geq30$ (here $n = 50$) and the population standard - deviation is unknown (we use the sample standard - deviation $s$), the formula is $\bar{x}\pm z_{\alpha/2}\frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard - deviation, $n$ is the sample size, and $z_{\alpha/2}$ for a 95% confidence interval is 1.96.
We are given $\bar{x}=4.2$ hours, $s = 1.7$ hours, and $n = 50$.

Step2: Calculate the margin of error $E$

The margin of error $E=z_{\alpha/2}\frac{s}{\sqrt{n}}$. Substitute $z_{\alpha/2}=1.96$, $s = 1.7$, and $n = 50$ into the formula.
$E=1.96\times\frac{1.7}{\sqrt{50}}\approx1.96\times\frac{1.7}{7.07}\approx1.96\times0.24=0.47$.

Step3: Calculate the lower and upper bounds of the confidence interval

The lower bound is $\bar{x}-E=4.2 - 0.47=3.73$ hours.
The upper bound is $\bar{x}+E=4.2 + 0.47=4.67$ hours.

Answer:

A. 3.73 to 4.67 hours