Homework

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, $\overline{X}$ = ____________
4. $\overline{x}$ ~ ______ (______, ______)
5. Find the probability that an individual had between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.
6. Find the probability that the average of the 25 students was between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.
7. Explain why there is a difference in part e and part f.

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, $\overline{X}$ = _____________
3. $\overline{X}$ ~ _____(_____,_____)
4. Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences.
5. Would you be surprised if one taxpayer finished their Form 1040 in more than 12 hours? In a complete sentence, explain why.

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, $\overline{X}$ = _____________
4. $\overline{X}$ ~ _____(_____,_____)
5. Find the first quartile for the average song length, $\overline{X}$.
6. The IQR (interquartile range) for the average song length, $\overline{X}$, is from ___ - ___.

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 $\overline{X}$ is approximately equal to the mean of Χ.
2. When the sample size is large, $\overline{x}$ is approximately normally distributed.
3. When the sample size is large, the standard deviation of $\overline{x}$ 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 words, $\overline{X}$ = _____________
3. $\overline{X}$ ~ _____(_____,_____)
4. How is it possible for the standard deviation to be greater than the average?
5. Why is it more likely that the average of the 1,000 residents will be from $2,000 to $2,100 than from $2,100 to $2,200?

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

Suppose that the duration of a particular type of criminal trial is known to have a mean of 21 days and a standard deviation of seven days. We randomly sample nine trials.

1. In words, ΣX = ______________
2. ΣX ~ _____(_____,_____)
3. Find the probability that the total length of the nine trials is at least 225 days.
4. Ninety percent of the total of nine of these types of trials will last at least how long?

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,500. We randomly survey ten teachers from that district.

1. In words, X = ______________
2. X ~ _____(_____,_____)
3. In words, ΣX = _____________
4. ΣX ~ _____(_____,_____)
5. Find the probability that the teachers earn a total of over $400,000.
6. Find the 90^th percentile for an individual teacher's salary.
7. Find the 90^th percentile for the sum of ten teachers' salary.
8. If we surveyed 70 teachers instead of ten, graphically, how would that change the distribution in part d?
9. If each of the 70 teachers received a $3,000 raise, graphically, how would that change the distribution in part b?

The closing stock prices of 35 U.S. semiconductor manufacturers are given as follows.

8.625; 30.25; 27.625; 46.75; 32.875; 18.25; 5; 0.125; 2.9375; 6.875; 28.25; 24.25; 21; 1.5; 30.25; 71; 43.5; 49.25; 2.5625; 31; 16.5; 9.5; 18.5; 18; 9; 10.5; 16.625; 1.25; 18; 12.87; 7; 12.875; 2.875; 60.25; 29.25

1. In words, Χ = ______________
2. 1. $\overline{x}$ = _____
   2. s_x = _____
   3. n = _____
3. Construct a histogram of the distribution of the averages. Start at x = –0.0005. Use bar widths of ten.
4. In words, describe the distribution of stock prices.
5. Randomly average five stock prices together. (Use a random number generator.) Continue averaging five pieces together until you have ten averages. List those ten averages.
6. Use the ten averages from part e to calculate the following.
   1. $\overline{x}$ = _____
   2. s_x = _____
7. Construct a histogram of the distribution of the averages. Start at x = -0.0005. Use bar widths of ten.
8. Does this histogram look like the graph in part c?
9. In one or two complete sentences, explain why the graphs either look the same or look different?
10. Based upon the theory of the central limit theorem, $\overline{X}$ ~ _____(_____,____)

The average wait time is:

1. one hour.
2. two hours.
3. two and a half hours.
4. four hours.

The 90^th percentile sample average wait time (in minutes) for a sample of 100 riders is:

1. 315.0
2. 40.3
3. 38.5
4. 65.2

What's the approximate probability that the average price for 16 gas stations is over $4.69?

1. almost zero
2. 0.1587
3. 0.0943
4. unknown

Suppose in a local Kindergarten through 12^th grade (K - 12) school district, 53 percent of the population favor a charter school for grades K through five. A simple random sample of 300 is surveyed. Calculate following using the normal approximation to the binomial distribtion.

1. Find the probability that less than 100 favor a charter school for grades K through 5.
2. Find the probability that 170 or more favor a charter school for grades K through 5.
3. Find the probability that no more than 140 favor a charter school for grades K through 5.
4. Find the probability that there are fewer than 130 that favor a charter school for grades K through 5.
5. Find the probability that exactly 150 favor a charter school for grades K through 5.

If you have access to an appropriate calculator or computer software, try calculating these probabilities using the technology.

X ~ N(60, 9). Suppose that you form random samples of 25 from this distribution. Let $\overline{X}$ be the random variable of averages. Let ΣX be the random variable of sums. For parts c through f, sketch the graph, shade the region, label and scale the horizontal axis for $\overline{X}$, and find the probability.

1. Sketch the distributions of X and $\overline{X}$ on the same graph.
2. $\overline{X}$ ~ _____(_____,_____)
3. P($\overline{x}$ < 60) = _____
4. Find the 30^th percentile for the mean.
5. P(56 < $\overline{x}$ < 62) = _____
6. P(18 < $\overline{x}$ < 58) = _____
7. Σx ~ _____(_____,_____)
8. Find the minimum value for the upper quartile for the sum.
9. P(1,400 < Σx < 1,550) = _____

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,500. We randomly survey ten teachers from that district.

1. Find the 90^th percentile for an individual teacher’s salary.
2. Find the 90^th percentile for the average teacher’s salary.

NeverReady batteries has engineered a newer, longer lasting AAA battery. The company claims this battery has an average life span of 17 hours with a standard deviation of 0.8 hours. Your statistics class questions this claim. As a class, you randomly select 30 batteries and find that the sample mean life span is 16.7 hours. If the process is working properly, what is the probability of getting a random sample of 30 batteries in which the sample mean lifetime is 16.7 hours or less? Is the company’s claim reasonable?

M&M candies large candy bags have a claimed net weight of 396.9 g. The standard deviation for the weight of the individual candies is 0.017 g. The following table is from a stats experiment conducted by a statistics class.

The bag contained 465 candies and he listed weights in the table came from randomly selected candies. Count the weights.

1. Find the mean sample weight and the standard deviation of the sample weights of candies in the table.
2. Find the sum of the sample weights in the table and the standard deviation of the sum of the weights.
3. If 465 M&Ms are randomly selected, find the probability that their weights sum to at least 396.9.
4. Is the Mars Company’s M&M labeling accurate?

Your company has a contract to perform preventive maintenance on thousands of air-conditioners in a large city. Based on service records from previous years, the time that a technician spends servicing a unit averages one hour with a standard deviation of one hour. In the coming week, your company will service a simple random sample of 70 units in the city. You plan to budget an average of 1.1 hours per technician to complete the work. Will this be enough time?

Certain coins have an average weight of 5.201 grams with a standard deviation of 0.065 g. If a vending machine is designed to accept coins whose weights range from 5.111 g to 5.291 g, what is the expected number of rejected coins when 280 randomly selected coins are inserted into the machine?