117 7.9 Topic 9: Sampling Distributions 71. In a few sentences, describe what the sampling distribution of the mean represents in your own 72. In 2012, MetricNet found that the average length of a phone call to a help desk was about ten minutes. Call times were strongly right skewed with a standard deviation around 3.5 minutes. Suppose we take a random sample of 50 calls to the help desk and find the average () call duration for the 50 calls. We repeat this process many times. (a) What do you expect the mean of the a's to be? (b) What do you expect the standard deviation of the 's to be? (c) What do you expect the shape of the a's to be? What guarantees this lac 73. The on-campus housing costs for a semester at a state university are Normally distributed with a mean of $1553 and a standard deviation of $329. (a) What is the distribution of the mean on-campus housing cost for a semester at this university for a random sample of 50 students? (b) What is the probability that a randomly selected on-campus student spends more than $1200 per semester? (C) What is the probability that a random sample of 50 students spends an average of less than $1500? dat is the probability that a random sample of 50 students spends an average of $1450 to $1600 per semester? 14. Sunshine City was designed to attract retired people, and its current population of 50,000 residents of 60 years ages is skewed has a mean age and a standard deviation of 16 years. The distribution of to the left. (a) What is the distribution of the sample mean age of residents for a random sample of 100 residents? (b) What is the probability that a random sample of 100 residents have a mean age between 50 and 70 years of age? (c) What is the probability that the mean age of a random sample of 100 residents is less than 55? 75. According to Quantcast analytics of CrossFit.com, 59% of CrossFitters have children. Suppose we take a random sample of 40 Crossfitters and find the sample proportion (p) that have children. We repeat this process many times. (a) What do you expect the mean of the p's to be? (b) What do you expect the standard deviation of the p's to be? (c) What do you expect the shape of the p's to be? 76. According to Quantcast analytics of CrossFit.com, 40% of CrossFitters have post-graduate de- grees. Suppose we take a random sample of 60 Crossfitters. (a) Describe the sampling distribution of the sample proportion of CrossFitters who have post- graduate degrees for samples of size 60. 118 (b) Find the probability that at least half of the sample 60 CrossFitters has post-graduate (c) Describe the sampling distribution of the sample proportion of CrossFitters who have post- (d) Find the probability that between 25% and 50% of the sample 100 CrossFitters has post- 77. At a university, 60% of the 7,400 students are female. The student newspaper reports results of degrees. graduate degrees for samples of size 100. graduate degrees. a survey of a random sample of 200 students about various topics involving alcohol abuse, such as participation in binge drinking. (a) Using the binomial distribution, find the probability that at most 104 students in a sample (b) Describe the sampling distribution of the sample proportion of females in samples of size (c) Suppose that out of a sample of 200 students, 52% are female (which is 104 out of 200). Find of 200 are female. 200 at this university. the probability that the sample proportion of females for a random sample of 200 students (d) Find the probability that between 50 and 60 percent of students in a random sample of 200 is at most 0.52. are female.