7.1 The Central Limit Theorem for Sample Means (Averages)

An unknown distribution has a mean of 90 and a standard deviation of 15. Samples of size n = 25 are drawn randomly from the population.

a. Find the probability that the sample mean is between 85 and 92.

b. Find the value that is two standard deviations above the expected value, 90, of the sample mean.

The length of time, in hours, it takes an "over 40" group of people to play one soccer match is normally distributed with a mean of two hours and a standard deviation of 0.5 hours. A sample of size n = 50 is drawn randomly from the population. Find the probability that the sample mean is between 1.8 hours and 2.3 hours.

In a recent study, it was reported that the mean age of iPad users is 34 years. Suppose the standard deviation is 15 years. Take a sample of size n = 100.

1. What are the mean and standard deviation for the sample mean ages of iPad users?
2. What does the distribution look like?
3. Find the probability that the sample mean age is more than 30 years (the reported mean age of iPad users in this particular s…

The mean number of minutes for app engagement by an iPad user is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of 60.

1. What are the mean and standard deviation for the sample mean number of minutes for app engagement by an iPad user?
2. What is the standard error of the mean?
3. Find the 90^th percentile for the sample mean time of minutes for app engagement by an iPad user. In…

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 revie…

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 revie…

Previously, De Anza statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of $0.88. Suppose that we randomly pick 25 daytime statistics students.

1. In words, Χ = ____________
2. Χ ~ _____(_____,_____)
3. In words, $\bar{X}$ = ____________
4. $\bar{x}$ ~ ______ (______, ______)
5. Find the probability that an individual had between $0.80 and $1.00. Gr…

According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is two hours. Suppose we randomly sample 36 taxpayers.

1. In words, Χ = _____________
2. In words, $\bar{X}$ = ________…

The length of songs in a collector’s Apple Music album collection is uniformly distributed from two to 3.5 minutes. Suppose we randomly pick five albums from the collection. There are a total of 43 songs on the five albums.

1. In words, Χ = _________
2. Χ ~ _____________
3. In words, $\bar{X}$ = _____________
4. $\bar{X}$ ~ _____(_____,_____)
5. Find the first quartile for the average song length, $\bar{X}$.
6. The IQR (interquartile rang…

Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.

1. When the sample size is large, the mean of $\bar{X}$ is approximately equal to the mean of Χ.
2. When the sample size is large, $\bar{x}$ is approximately normally distributed.
3. When the sample size is large, the standard deviation of $\bar{x}$ is approximately the same as the standard deviation of Χ.

The distribution of income in some developing countries is considered wedge shaped (many low income people, very few middle income people, and even fewer high income people). Suppose we pick a country with a wedge shaped distribution. Let the average salary be $2,000 per year with a standard deviation of $8,000. We randomly survey 1,000 residents of that country.

1. In words, Χ = _____________
2. In wo…

The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. The distribution to use for the average cost of gasoline for the 16 gas stations is:

1. $\bar{X}$ ~ N(4.59, 0.10)
2. $\bar{X}$ ~ N$\left(\text{4} .\text{59},\textrm{ } \frac{0.10}{\sqrt{16}}\right)$
3. $\bar{X}$ ~ N$\left(\text{4} .\text{59},\textrm{ } \frac{16}{0.10}\right)$
4. $\bar{X}$ ~ N$\left(\text{4} .\text{59},\textrm{ } \frac{\sqrt{16}}{0.10}\right)$