7.1 Problem Set

1) Consider the histogram shown below.

[Figure12]

a) Make a sketch of the histogram and overlay a sketch of a density curve for the histogram.

b) What is the area under your density curve?

c) What is the shape of the density curve?

2) A roadside bait salesman digs up worms to sell to fishermen.

It turns out that the worms have a mean length of = 112 mm and a standard deviation of = 12 mm.

$\mu$

(Greek letter mu) for mean

$\sigma$

(Greek letter sigma ) for standard deviation

a) Draw and label a normal curve for this distribution. Include lines for the mean and for 1, 2, and 3 standard deviations above and below the mean.

b) What percentage of the worms will have lengths longer than 112 mm?

c) What percentage of the worms will have lengths between 100 and 124 mm long?

d) What percentage of the worms will have lengths between 100 and 112 mm long?

e) What percentage of the worms are longer than 124 mm? [Figure13]

f) What percentage of the worms are shorter than 88 mm?

3) Sketch a normal curve which has a mean of 13 pounds and a standard deviation of 3 pounds. Include lines for the mean and for 1, 2, and 3 standard deviations above and below the mean.

4) Not all 12-ounce cans of soda are the same. It turns out that the average 12-ounce can of soda does contain twelve ounces of soda, but the amount of soda is normally distributed with a standard deviation of 0.15 ounces. Fill in the blanks for each statement below.

a) The middle 68% of all 12-ounce soda cans contain between ____ & ____ ounces of soda.

b) The middle 95% of all 12-ounce soda cans contain between ____ & ____ ounces of soda.

c) The middle 99.7% of all 12-ounce soda cans contain between ____ & ____ ounces of soda.

5) Figure 7.2 on the following page shows an approximate distribution of the number of fish caught by the competitors during a one hour pan-fishing contest. Give the approximate values of the mean and the standard deviation for the distribution.

[Figure14]

6) Suppose the weights of adult males of a particular species of whale are distributed normally with a mean of 11,600 pounds and a standard deviation of 640 pounds.

a) Draw a normal curve for this situation. Use vertical lines to mark and label the mean and 1, 2, and 3 standard deviations above and below the mean.

b) What percent of these whales weigh less than 10,320 pounds?

c) Between what two weights do the middle 99.7% of these whales weigh?

d) What percent of these whales weigh between 10,320 pounds and 12,240 pounds?

7) Which situation is most likely to be normally distributed? Explain your reasoning.

i) The hair lengths for all the Statistics and Probability students who have Mr. Johnson as a teacher.

ii) The prices of all new Ipod Touches that are sold in Minnesota this week.

iii) The average running times for all 4th grade boys at Andover Elementary in the 50 yard dash.

8) Suppose a standard incandescent light bulb will run an average of 400 hours before burning out. Of course, some bulbs burn out sooner and some last longer. Suppose that the average lives of these bulbs is normally distributed with a standard deviation of 35 hours.

a) Sketch and label a normal curve to illustrate this situation.

b) What percent of these bulbs will burn out in 400 hours or less?

c) If you are lucky, your bulb will last longer than advertised. What percent of bulbs should last 435 hours or more? What percent of bulbs will last 470 hours or more?

d) If you had 5000 bulbs that you needed for use in a large office building, how many would you expect to last at least 365 hours?

9) Suppose that the time that it takes for a popcorn kernel to pop produces a normal distribution with a mean of 145 seconds and a standard deviation of 13 seconds for a standard microwave oven.

a) It is usually not a good idea to let the microwave oven run until all the kernels are popped because some of the popcorn will start to burn. Suppose the ideal time to shut off the microwave oven is after about 97.5% of the kernels have popped. When will 97.5% of the kernels be popped?

b) Between what two times will we see the middle 68% of kernels popped?

10) After a great deal of surveying, it is determined that the average wait times in the cafeteria line are normally distributed with a mean of 7 minutes and a standard deviation of 2 minutes. Suppose that 400 students are released to the cafeteria for 2nd lunch.

a) Approximately how many students will have to wait more than 5 minutes for their food?

b) Approximately how many students will have to wait more than 11 minutes for their food?

11) Sudoku is a popular logic game of number combinations. It originated in the late 1800s in the French press, Le Siècle. The mean time it takes the average 11th grader to complete the Sudoku puzzle on the following page was found to be 19.2 minutes, with a standard deviation of 3.1 minutes.

a) Draw a normal distribution curve to represent this data.

b) Suppose Andover High School is going to put together a Sudoku team. The coach has decided that she will only consider players who score in the fastest 2.5% of the junior class as she puts together the team. How fast must a student solve a puzzle to be in the top 2.5% of puzzle solvers?

c) If there are 400 kids in the Andover junior class, how many of them will be able to solve the Sudoku puzzle below in 16.1 minutes or less?

12) In order to qualify for undercover detective training, a police officer must take a stress tolerance test. Scores on this test are normally distributed with a mean of 60 and a standard deviation of 10. Only the top 16% of police officers score high enough on the test to qualify for the detective training. What is the cutoff score that marks the top 16% of all scores?

Review Exercises

13) Use your calculator to find the mean and standard deviation of the data set below.

3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8

14) A pet store must select 2 dogs and 2 cats for display in their front window. In how many ways can this be done if there are 16 dogs and 12 cats available to choose from?

[Figure17]

15) By hand, give the five number summary for the data set below.

3, 5, 5, 6, 8, 9, 10, 10, 12, 13, 13, 13, 14, 15, 17, 19, 19, 20

16) A student conducts a survey in which 100 tenth-graders are asked "What is your favorite item on the lunch menu at school today?" The student decides to conduct this survey by handing each tenth-grader a survey sheet while they are eating and asking them to fill it out and turn it in to room P202 by the end of the day. Why will this survey method have a problem with bias?

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