1

Day #6: Chapter 7 Test Review I can distinguish between a parameter and a statistic.

1. For each description below, identify each underlined number as a parameter or statistic. Use appropriate notation to describe each number, e.g., ̂ = 0.96 . (a) A 1993 survey conducted by the Richmond Times-Dispatch one week before election day asked voters which candidate for the state’s attorney general they would vote for. 37% of the respondents said they would vote for the Democratic candidate. On election day, 41% actually voted for the Democratic candidate.

(b) The National Center for Health Statistics reports that the mean systolic blood pressure for males 35 to 44 years of age is 128 and the standard deviation is 15. The medical director of a large company looks at the medical records of 72 executives in this age group and finds that the mean systolic blood pressure for these executives is 126.07.

I can determine whether a statistic is an unbiased estimator of a population parameter.

I can understand the relationship between sample size and the variability of an estimator.

2. Suppose two different statistics—call them Statistic A and Statistic B—can be used to estimate the same population parameter. Statistics A has lower bias than B, but A also has high variability compared to B. On the two axes below, draw two parallel dotplots showing 8 values of each statistic that are consistent with these characteristics. Assume that the parameter value is at the arrow on the axes.

2

I can understand the definition of a sampling distribution.

I can distinguish between population distribution, sampling distribution, and the distribution of sample data.

3. A large pet store that specializes in tropical fish has several thousand guppies. The store claims that the guppies have a mean length of 5 cm and a standard deviation of 0.5 cm. You come to the store and buy 10 randomly-selected guppies and find that the mean length of your 10 guppies is 4.8 cm. This makes you suspect that the mean fish length is not what the store says it is. To explore this further, you assume that the length of guppies is Normally distributed and use a computer to simulate 200 samples of 10 guppies from the store’s claimed population. Below is a dotplot of the means from these 200 samples.

(a) What is the population in this situation, and what population parameters have we been given? (b) The distribution of one sample is described in the opening paragraph. What information have we been given about this sample? (c) Is the dotplot above a sampling distribution? Explain. (d) Do you think the store is being honest about the length of its guppies? Justify your answer.

3

I can find the mean and standard deviation of the sampling distribution of a sample proportion p̂ for an SRS of size n from a population having

proportion p of successes. I can check whether the 10% and Normal conditions are met in a given

setting. I can use Normal approximation to calculate probabilities involving p̂ .

I can use the sampling distribution of p̂ to evaluate a claim about a

population proportion. 4. According to the 2000 U.S. Census, 80% of Americans over the age of 25 have earned a high school diploma. Suppose we take a random sample of 120 Americans and record the proportion, ̂, of individuals in our sample that have a high school diploma. (a) What are the mean and standard deviation of the sampling distribution of ̂ ? (b) What is the approximate shape of the sampling distribution? Justify your answer. (c) Suppose our sample size was 30 instead of 120. Compare the shape, center, and spread of this sampling distribution to the one in parts (a) and (b).

4 (d) You live in a small town with only 500 residents over the age of 25. What is the largest possible sample you can take from your town and still be able to calculate the standard deviation of sampling distribution of ̂ using the method presented in the textbook? Explain. 5. George is a big fan of music from the 1960s, and 22% of the songs on his mp3 player are Beatles songs. Suppose George sets his mp3 player to “shuffle,” so that it selects songs randomly (assume the shuffle function permits repetition of songs). During a long drive, George plays 50 randomly-selected songs. (a) What are the mean and standard deviation of the proportion of the 50 randomly-selected songs that are Beatles songs? (b) Calculate the probability that more than 30% of the 50 randomly-selected songs are Beatles songs.

5

I can find the mean and standard deviation of the sampling distribution of a sample mean x from an SRS of size n.

I can calculate probabilities involving a sample mean x when the population distribution is Normal.

I can explain how the shape of the sampling distribution of x is related to the shape of the population distribution.

I can use the central limit theorem to help find probabilities involving a sample mean x .

6. The customer care manager at a cell phone company keeps track of how long each help-line caller spends on hold before speaking to a customer service representative. He finds that the distribution of wait times for all callers has a mean of 12 minutes with a standard deviation of 5 minutes. The distribution is moderately skewed to the right. Suppose the manager takes a random sample of 10 callers and calculates their mean wait time, x . (a) What is the mean of the sampling distribution of x ? (b) Is it possible to calculate the standard deviation of x ? If it is, do the calculation. If it isn’t, explain why. (c) Do you know the approximate shape of the sampling distribution of x ? If so, describe the shape and justify your answer. If not, explain why not.

6 7. The weights of Granny Smith apples from a large orchard are Normally distributed with a mean of 380 gm and a standard deviation of 28 gm. (a) A single apple is selected at random from this orchard. What is the probability that it weighs more 400 gm? (b) Three apples are selected at random from this orchard. What is the probability that their mean weight is greater than 400 gm.? (c) Explain why the probabilities in (a) and (b) are not equal.

7

Day #7: Chapter 7 Test Review FRAPPY #1: IPOD’s a. David’s iPod has about 10,000 songs. The distribution of the

play times for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. Suppose we choose an SRS of 10 songs from this population and calculate the mean play time ̅ of these songs. What are the mean and standard deviation of the sampling distributions of ̅? Explain.

b. Explain why you cannot safely calculate the probability that the mean play time ̅ is more than 4 minutes (240 seconds) for an SRS of 10 songs.

c. Suppose that we take an SRS of 36 songs instead. Explain how the central limit theorem allows us to find the probability that the mean play time is more than 240 seconds. Then calculate this probability. Show your work.

8

FRAPPY #2: ‘Bottling Cola’: A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 millimeters (mL). In fact, the contents vary according to a normal distribution with mean μ = 298 mL and standard deviation σ = 3 mL. a. What is the probability that an individual bottle contains less

than 295 mL?

b. What is the probability that the mean contents of the bottle in a six-pack is less than 295 mL?

c. What is the probability that the mean contents of the bottle in a six-pack is greater than 299 mL?

9

FRAPPY #3: ‘Polling Women’: Suppose that 47% of all adult women think they do not get enough time for themselves. An opinion poll interviews 1025 randomly chosen women and records the sample proportion who feels they don’t get enough time for themselves. a. Describe the sampling distribution of ̂.

b. The truth about the population is p = 0.47. In what range will the middle 95% of all sample results fall?

c. What is the probability that the poll gets a sample in which fewer than 45% say they do not get enough time for themselves?

10

FRAPPY #4: “Candy Bars” The distribution of actual weights of 8-ounce chocolate bars produced by a certain machine is Normal with mean 8.1 ounces and standard deviation 0.1 ounces. Company managers do not want the weight of a chocolate bar to fall below 7.85 ounces, for fear that consumers will complain.

a. Find the probability that the weight of a randomly selected candy bar is less than 7.85 ounces.

b. Four candy bars are selected at random and their mean weight, x , is computed. Describe the center, shape, and spread of the sampling distribution of x .

c. Find the probability that the mean weight of the four candy bars is less than 7.85 ounces.

11

Day #8: Chapter 7 Test Review FRAPPY #5: ELECTRICAL PROBLEMS: It is generally believed that electrical problems affect about 14% of all new cars. An automobile mechanic conducts diagnostic tests on 128 new cars on the lot. a. Describe the sampling distribution for the sample proportion by telling

its mean and standard deviation. Justify your answer. (Be sure to check for normality and to address any conditions/assumptions)

b. What is the probability that in this group of new cars, over 18% will be found to have electrical problems?

c. What is the probability that in this group of new cars, between 11% and 17% will be found to have electrical problems?

12

FRAPPY #6: SALES TAX: A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let p denote the proportion in the sample that says they support the increase. Suppose that 40% of all adults in Ohio support the increase. a. If ̂ is the proportion of the sample who support the increase, what is

the mean of the sampling distribution of ̂?

b. What is the standard deviation of ̂?

c. Explain why you can use the formula for the standard deviation of ̂ in this setting.

d. Check that you can use the normal approximation for the distribution of ̂.

13

FRAPPY #7: SODA: A certain beverage company is suspected of under filling its cans of soft drink. The company advertises that its cans contain, on the average, 12 ounces of soda with standard deviation 0.4 ounce. a. Compute the probability that a random sample of 50 cans produces a

sample mean fill of 11.9 ounces or less. (A sketch of the distribution is required.)

b. Suppose that each of the 25 students in a statistics class collects a random sample of 50 cans and calculates the mean number of ounces of soda. Describe the approximate shape of the distribution for these 25 values of

x .

c. What important principle that we studied is used to answer the previous question?

14

FRAPPY #8: Television (from Strive Book): A television producer must schedule a selection of paid advertisements during each hour of programming. The lengths of the advertisements are Normally distributed with a mean of 28 seconds and standard deviation of 5 seconds. During each hour of programming, 45 minutes are devoted to the program and 15 minutes are set aside for advertisements. To fill in the 15 minutes, the producer randomly selects 30 advertisements. a. Describe the sampling distribution of the sample mean length for the random samples of 30 advertisements. b. If 30 advertisements are randomly selected, what is the probability that the total time needed to air them will exceed the 15 minutes available? Show your work.