Practice
7.1 The Central Limit Theorem for Sample Means (Averages)
Use the following information to answer the next six exercises: Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let $\overline{X}$ be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.
What is the mean, standard deviation, and sample size?
Complete the distributions.
 X ~ _____(_____, _____)
 $\overline{X}$ ~ _____(_____, _____)
Find the probability that one review will take Yoonie from 3.5 to 4.25 hours. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

 P(________ x ________) = _______
Find the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

 P(________________) = _______
Find the 95^{th} percentile for the mean time to complete one month's reviews. Sketch the graph.

 The 95^{th} percentile =____________
7.2 The Central Limit Theorem for Sums (Optional)
Use the following information to answer the next four exercises: An unknown distribution has a mean of 80 and a standard deviation of 12. A sample size of 95 is drawn randomly from the population.
Find the probability that the sum of the 95 values is greater than 7,650.
Find the probability that the sum of the 95 values is less than 7,400.
Find the sum that is two standard deviations above the mean of the sums.
Find the sum that is 1.5 standard deviations below the mean of the sums.
Find the probability that the sum of the 40 values is greater than 7,500.
Find the probability that the sum of the 40 values is less than 7,000.
Find the sum that is one standard deviation above the mean of the sums.
Find the sum that is 1.5 standard deviations below the mean of the sums.
Find the percentage of sums between 1.5 standard deviations below the mean of the sums and one standard deviation above the mean of the sums.
Find the probability that the sum of the 100 values is greater than 3,910.
Find the probability that the sum of the 100 values is less than 3,900.
Find the probability that the sum of the 100 values falls between the numbers you found in Exercise 7.16 and Exercise 7.17.
Find the sum with a zscore of –2.5.
Find the sum with a zscore of 0.5.
Find the probability that the sums will fall between the zscores –2 and 1.
What is the mean of ΣX?
What is the standard deviation of ΣX?
What is P(Σx = 290)?
What is P(Σx > 290)?
True or False: Only the sums of normal distributions are also normal distributions.
In order for the sums of a distribution to approach a normal distribution, what must be true?
What three things must you know about a distribution to find the probability of sums?
An unknown distribution has a mean of 25 and a standard deviation of six. Let X = one object from this distribution. What is the sample size if the standard deviation of ΣX is 42?
An unknown distribution has a mean of 19 and a standard deviation of 20. Let X = the object of interest. What is the sample size if the mean of ΣX is 15,200?
What is the zscore for Σx = 840?
What is the zscore for Σx = 1,186?
What is P(Σx 1,186)?
What is the mean of ΣX?
What is the standard deviation of ΣX?
What is P(Σx > 9000)?
7.3 Using the Central Limit Theorem
Use the following information to answer the next 10 exercises: A manufacturer produces 25pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.
 What is the distribution for the weights of one 25pound lifting weight? What are the mean and standard deviation?
 What is the distribution for the mean weight of 100 25pound lifting weights?
 Find the probability that the mean actual weight for the 100 weights is less than 24.9.
Find the probability that the mean actual weight for the 100 weights is greater than 25.2.
Find the 90^{th} percentile for the mean weight for the 100 weights.
 What is the distribution for the sum of the weights of 100 25pound lifting weights?
 Find P(Σx 2450).
Find the 90^{th} percentile for the total weight of the 100 weights.
 What is the standard deviation?
 What is the parameter m?
What is the distribution for the length of time one battery lasts?
What is the distribution for the mean length of time 64 batteries last?
What is the distribution for the total length of time 64 batteries last?
Find the probability that the sample mean is between 7 and 11.
Find the 80^{th} percentile for the total length of time 64 batteries last.
Find the interquartile range (IQR) for the mean amount of time 64 batteries last.
Find the middle 80 percent for the total amount of time 64 batteries last.
Find P(Σx > 420).
Find the 90^{th} percentile for the sums.
Find the 15^{th} percentile for the sums.
Find the first quartile for the sums.
Find the third quartile for the sums.
Find the 80^{th} percentile for the sums.