The amount of time adults spend watching television is closely monitored by firms because this helps to determine advertising pricing for commercials. Complete parts (a) through (d). (a) Do you think the variable weekly time spent watching television" would be normally distributed? If not, what shape would you expect the variable to have? A. The variable "weekly time spent watching television" is likely symmetric, but not normally distributed. B. The variable "weekly time spent watching television" is likely uniform, not normally distributed. C. The variable "weekly time spent watching television" is likely skewed left, not normally distributed. D. The variable "weekly time spent watching television" is likely skewed right, not normally distributed. E. The variable "weekly time spent watching television" is likely normally distributed. (b) According to a certain survey, adults spend 2.45 hours per day watching television on a weekday. Assume that the standard deviation for "time spent watching television on a weekday is 1.93 hours. If a random sample of 50 adults is obtained, describe the sampling distribution of $\bar{x}$, the mean amount of time spent watching television on a weekday. $\tilde{x}$ is approximately normal with $\mu_{x}=2.45$ and $\sigma_{x}=0.272943$. (Round to six decimal places as needed) (c) Determine the probability that a random sample of 50 adults results in a mean time watching television on a weekday of between 2 and 3 hours. The probability is $\square$. (Round to four decimal places as needed)
Solution
Step 1 ：The variable 'weekly time spent watching television' is likely skewed right, not normally distributed.
Step 2 ：The sampling distribution of the mean time spent watching television on a weekday for a sample of 50 adults is approximately normal with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the mean is \(2.45\) hours and the standard deviation is approximately \(0.273\) hours.
Step 3 ：To find the probability that a random sample of 50 adults results in a mean time watching television on a weekday of between 2 and 3 hours, we standardize the values 2 and 3 by subtracting the mean and dividing by the standard deviation to get z-scores. The z-scores are approximately \(-1.65\) and \(2.02\).
Step 4 ：We then use a z-table or a function that gives the cumulative distribution function of the standard normal distribution to find the probabilities corresponding to these z-scores. The probabilities are approximately \(0.0496\) and \(0.978\).
Step 5 ：The probability that the mean is between 2 and 3 hours is the difference between these two probabilities, which is approximately \(0.9284\).
Step 6 ：Final Answer: The probability that a random sample of 50 adults results in a mean time watching television on a weekday of between 2 and 3 hours is approximately \(\boxed{0.9284}\).
