91Ó°ÊÓ

Cans of a cola beverage claim to contain 16 ounces. The amounts in a sample are measured and the statistics are $n=34,\stackrel{¯}{x}=16.01$ounces. If the cans are filled so that $\mathrm{μ}=16.00$ounces (as labeled) and $\mathrm{σ}=0.143$ounces, find the probability that a sample of 34 cans will have an average amount greater than $16.01$ounces. Do the results suggest that cans are filled with an amount greater than 16 ounces?

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

a. In words,$X=\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

b.$X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

c. In words,$\mathrm{Î\pounds }X=\_\_\_\_\_\_\_\_\_\_\_\_\_$

d.$\mathrm{Î\pounds }X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

e. Find the probability that the teachers earn a total of over $\backslash (400,000$.

f. Find the $90th$percentile for an individual teacher's salary.

g. Find the $90th$percentile for the sum of ten teachers' salary.

h. If we surveyed $70$teachers instead of ten, graphically, how would that change the distribution in part d?

i. If each of the $70$teachers received a $\backslash )3,000$raise, graphically, how would that change the distribution in part b?

An unknown distribution has a mean of 45 and a standard deviation of eight. A sample size of 50 is drawn randomly from the population. Find the probability that the sum of the 50 values is more than $2,400.$

76. The attention span of a two-year-old is exponentially distributed with a mean of about eight minutes. Suppose we randomly survey 60 two-year-olds.

a. In words, X=

b. $X~$

c. In words$\stackrel{-}{X}$=

d. $\stackrel{-}{X}~$

e. Before doing any calculations, which do you think will be higher? Explain why.

i. The probability that an individual attention span is less than ten minutes.

ii. The probability that the average attention span for the 60 children in less than ten minutes?

f. Calculate the probabilities in part e.

g. Explain why the distribution for $\stackrel{-}{X}$ is not exponential.

In a recent study reported Oct.29, 2012 on the Flurry Blog, the mean age of tablet users is 35 years. Suppose the standard deviation is ten years. The sample size is 39 .

a. What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution?

b. Find the probability that the sum of the ages is between 1,400 and 1,500 years.

c. Find the ${90}^{\text{th}}$ percentile for the sum of the 39 ages.

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

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

a. In words,$\mathrm{Î\S }=\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

b. i.$\overline{x}=\_\_\_\_\_$

ii.$s_x=\_\_\_\_\_$

iii.$n=\_\_\_\_\_$

c. Construct a histogram of the distribution of the averages. Start at $x=–0.0005$. Use bar widths of ten.

d. In words, describe the distribution of stock prices.

e. 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.

f. Use the ten averages from part e to calculate the following.

i.$\overline{x}=\_\_\_\_\_$

ii.$s_x=\_\_\_\_\_$

g. Construct a histogram of the distribution of the averages. Start at $x=-0.0005$. Use bar widths of ten.

h. Does this histogram look like the graph in part c?

i. In one or two complete sentences, explain why the graphs either look the same or look different?

j. Based upon the theory of the central limit theorem,$X¯~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_)$

Use the information in Example 7.8, but use a sample size of 55 to answer the following questions. a. Find P( x ¯ < 7).

b. Find P(Σx > 170).

c. Find the 80th percentile for the mean of 55 scores.

d. Find the 85th percentile for the sum of 55 scores.

Use the information in Example \(7.9\), but change the sample size to \(144\).

a. Find \(P(20<\bar{x}<30)\).

b. Find \(P(\sum x\) is at least \(3,000)\).

c. Find the \(75th\) percentile for the sample mean excess time of \(144\) customers.

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 less than $7400$.

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

a. $315.0$

b.$40.3$

c.$38.5$

d.$65.2$