The number of hours per week that high school seniors spend on homework is normally distributed, with a mean of 10 hours and a standard deviation of 3 hours.60 students are chosen at random.Let represent the mean number of hours spent on homework for this group.Find the probability that is between 9.8 and 10.4.
A)0.547
B)0.5161
C)0.080
D)0.1528
E)0.3043

Question

Quizplus · Accepted Answer

The sampling distribution of the sample mean for a sample size of 60, given the population is normally distributed, will also be normally distributed with a mean ($\mu_{\bar{x}}$) equal to the population mean ($\mu$) and a standard deviation ($\sigma_{\bar{x}}$) equal to the population standard deviation ($\sigma$) divided by the square root of the sample size ($n$), i.e., $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$. Here, $\mu = 10$ hours, $\sigma = 3$ hours, and $n = 60$. Thus, $\sigma_{\bar{x}} = \frac{3}{\sqrt{60}}$.To find the probability that $\bar{x}$ is between 9.8 and 10.4, we convert these values to Z-scores using the formula $Z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$.For 9.8: $Z = \frac{9.8 - 10}{3/\sqrt{60}}$For 10.4: $Z = \frac{10.4 - 10}{3/\sqrt{60}}$Calculating these Z-scores and then using the standard normal distribution table or a calculator, we find the probabilities corresponding to these Z-scores and subtract the smaller from the larger to find the probability that $\bar{x}$ is between 9.8 and 10.4. This calculation yields a probability of approximately 0.547, making choice A correct.

The Number of Hours Per Week That High School Seniors