7.1 The Central Limit Theorem for Sample Means

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, $\overset{–}{X}$ = ____________
4. $\overset{–}{X}$ ~ ______ (______, ______)
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, $\overset{–}{X}$ = ________…

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, $\overset{–}{X}$ = _____________
4. $\overset{–}{X}$ ~ _____(_____,_____)
5. Find the first quartile for the average song length.
6. The IQR(interquartile range) f…

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 $\overset{–}{X}$ is approximately equal to the mean of Χ.
2. When the sample size is large, $\overset{–}{X}$ is approximately normally distributed.
3. When the sample size is large, the standard deviation of $\overset{–}{X}$ 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. $\overset{–}{X}$ ~ N(4.59, 0.10)
2. $\overset{–}{X}$ ~ N$\left(\text{4} .\text{59},\textrm{ } \frac{0.10}{\sqrt{16}}\right)$
3. $\overset{–}{X}$ ~ N$\left(\text{4} .\text{59},\textrm{ } \frac{16}{0.10}\right)$
4. $\overset{–}{X}$ ~ N$\left(\text{4} .\text{59},\textrm{ } \frac{\sqrt{16}}{0.10}\right)$