7.1.1 Calculate the mean and standard deviation for the sampling distribution of a sample mean.

7.1.2 Calculate probabilities for the sampling distribution of a sample mean using technology.

7.1.3 Calculate percentiles for the sampling distribution of a sample mean using technology.

7.3.1 Use the central limit theorem to calculate probabilities for normal and non-normal population distributions.

7.3.2 Use the central limit theorem to calculate percentiles for normal and non-normal population distributions.

Central Limit Theorem

1. In what way is the central limit theorem similar to the law of large numbers?

1. Using the same population, which sampling distribution for a sample mean would have more variability: a sampling distribution based on a sample size of n = 15 or a sampling distribution based on a sample size of n = 25?

1. Settle this debate. Luca says that if the sample size is doubled the standard deviation of the sampling distribution is cut in half. Carson says that if the sample size is quadrupled the standard deviation is cut in half. Who is right? Support your answer with appropriate calculations.

1. Findmysuitcase.com delivers suitcases that have been lost on airlines to their owners. However, they cannot specify an exact delivery time due to the number of suitcases and locations for the deliveries. People are told that the delivery will be in a 4 hour window. Assume that the distribution of delivery times is uniformly distributed and X = the number of minutes spent waiting in the 4 hour window.

1. What is the mean wait time?

1. What is the standard deviation of wait times?

1. What is the probability your suitcase will be delivered in the first 3 hours of the 4 hour window?

1. Findmysuitcase.com has received many complaints about slow deliveries. They select a random sample of 100 wait times. What is the probability that the mean wait time exceeds 3 hours?

1. If a new random sample of 100 wait times had a mean weight time of more than 3 hours, what would you recommend to the company?

1. What is the percentile associated with a mean wait time of 30 minutes for an individual customer?

1. Commuting times tend to follow a skewed distribution. Data on commuting times and distances for a sample of 500 people in Atlanta was gathered from the U.S. Census Bureau's American Housing Survey. This data was gathered in 2019.

1. Do you think the distribution of commuting times would be skewed to the left or skewed to the right? Explain your choice.

1. Is a sample size of 500 large enough to draw a conclusion about the shape of the sampling distribution for the sample mean? Explain.

1. You think that commuting times over the past two years (2020-2021) are less than they were before. Prior to 2020, the mean commuting time for all Atlanta commuters was 29 minutes with a standard deviation of 25 minutes. A random sample of 30 commuters has a mean time of 15 minutes for their commute. What is the probability that a random sample of 30 commuters would have a mean commute time of less than 15 minutes, if the distribution of commute times was unchanged?

1. Based on your answer to the previous question, do you think the distribution of commuting times has changed?

1. According to the website nurse.org, the highest average salary for nurses is in California at $54.44 per hour. Assume the distribution of nurses salaries in California are normally distributed.

1. Which would be more likely: a random sample of 15 nurses having a mean salary that is greater than $45 or a random sample of 50 nurses having a mean salary that is greater than $45?

1. Suppose the standard deviation of the sampling distribution of the sample mean salary for random samples of size 50 is $2.25. If the mean salary is $54.44, use the normal distribution to compute the probability that a random sample of 50 nurses will have a mean salary that is less than $50.

1. Suppose the distribution of salaries was non-normal but had the same mean and standard deviation. Would it still be appropriate to use the normal distribution to compute the probability in part b? Explain