Name_____________________ Chapter 9.1 A Homework
Done on Time:____/5 Completed:____/5
Key Vocabulary:
 parameter
 statistic
 sampling variability
 sampling distribution
 unbiased
Calculator Skills:
 randNorm(μ, σ, #trials )
9.1 Sampling Distributions (pp.562-580)
1. Explain the difference between a parameter and a statistic ?
A parameter is a measurement of the entire population. A statistics is an estimate of the population parameter obtained from a sample of the population.
2. What is sampling variability ?
Sampling variability is what happens when we take several samples from the same population.
Every time we get a sample and find a statistic, it changes for each sample, or varies.
3. Explain the difference between 𝜇 𝑥
and 𝑥̅ , and between p and 𝑝̂ ?
The first one is for the population mean and the second for the sample mean. P is for the population proportion and the p hat is for the sample proportion
4. What is meant by the sampling distribution of a statistic?
The sampling distribution of a statistic is the distribution we get when we collect a sample, calculate the statistic, collect another sample, calculate the statistic, and then create a histogram from these statistics. The calculated statistics from ALL possible combinations of sample of size n create the sampling distribution.

5. When is a statistic considered unbiased ?
When its sampling distribution is not centered around the true population parameter.
6. How is the size of a sample related to the spread of the sampling distribution?
The larger your sample size the smaller the spread of the sampling distribution.
For each boldfaced number in number 7 and 8 give the following: (1) state whether it is a parameter or a statistic and (2) use appropriate notation to describe each number; for example, p = 0.65.
7. (a) The ball bearings in a large container have mean diameter 2.5003 centimeters (cm). This is within the specifications for acceptance of the container by the purchaser. By chance, an inspector chooses 100 bearings from the container that have mean diameter 2.5009 cm . Because this is outside the specified limits, the container is mistakenly rejected.
(b) The Bureau of Labor Statistics last month interviewed 60,000 members of the U.S. labor force, of whom 7.2% were unemployed.
(a) μ =2.5003 is a parameter (related to the population of all the ball bearings in the container and x =2.5009 is a statistic (related to the sample of 100 ball bearings). (b) ˆ p =7.2% is a statistic
(related to the sample of registered voters who were unemployed).
8. (a) A telemarketer firm in Los Angeles uses a device that dials residential telephone numbers in that city at random. Of the first 100 numbers dialed, 48% are unlisted. This is not surprising, because 52% of all Los Angeles residential phones are unlisted.
(b) A researcher carries out a randomized comparative experiment with young rats to investigate the effects of a toxic compound in food. She feeds the control group a normal diet. The experimental group receives a diet with 2500 parts per million of the toxic material. After 8 weeks, the mean weight gain is 355 grams for the control group and 289 grams for the experimental group.
(a) ˆ p = 48% is a statistic; = 52% is a parameter. (b) Both p control x = 335 and experimental x = 289 are statistics.

9. If a piece of toast falls off your breakfast plate, is it more likely to land with the buttered side down? According to Murphy’s Law (the assumption that if anything can go wrong, it will) the answer is “Yes.” Most scientists would argue that be the laws of probability, the toast is equally likely to land butter-side up or butter-side down. Robert Mathews, science correspondent of the
Sunday Telegraph, disagrees. He claims that when toast falls off a plate that is being carried at
“typical height,” the toast has just enough time to rotate once (landing butter-side down) before it lands. To test his claim, Mr. Mathews has arranged for 150,000 students in Great Britain to carry out an experiment with tumbling toast.
Assuming scientists are correct, the proportion of times that the toast will land butter-side down is p=0.5. We can use a coin to simulate the experiment. Let heads represent the toast landing butter-side down.
(a) Toss a coin 20 times and record the proportion of heads obtained, 𝑝̂ = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠
.
20
Explain how your result relates to the tumbling-toast experiment.
(b) Repeat this sampling process 10 times. Make a histogram of the 10 values of 𝑝̂ . Is the center of the distribution close to 0.5?
(c) Ten repetitions give a very crude approximation to the sampling distribution. Pool your work with that of other students to obtain several hundred repetitions. Make a histogram of all the values of 𝑝̂ . Is the center close to point 5? Is the shape approximately Normal?
(d) How much sampling variability is present? That is, how much do your values of 𝑝̂ based on sample sized of 20 differ from the actual population proportion, p=0.5?
(e) Why do you think Mr. Mathews is asking so many students to participate in his study?

(a) Since the proportion of times the toast will land butter-side down is 0.5, the result of 20 coin flips will simulate the outcomes of 20 pieces of falling toast (landing butter-side up or butter-side down).
(b) Answers will vary. A histogram for one simulation is shown below (on the left). The center of the distribution is close to 0.5. (c) Answers will vary. A histogram based on pooling the work of 25 students (250 simulated values of ˆ p ) is shown below (on the right). As expected, the simulated distribution of ˆ p is approximately Normal with a center at 0.5.
(d) Answers will vary, but the standard deviation will be close to 0.50.50.111820×  . The simulation above for the pooled results for 25 students produced a standard deviation of 0.1072.
(e) By combining the results from many students, he can get a more accurate estimate of the value of p since the value of ˆ p approaches p as the sample size increases.
10. Use your calculator to replicate problem 10 as follows:
The command randBin(20, .5) simulates tossing a coin 20 times. The output is the number of heads in 20 tosses. The common randBin(20, .5, 10)/20 simulates 10 repetitions of tossing coin
20 times and finding the proportion of heads. Go into your Statistics/List Editor and place your cursor on the top of L
1
/list 1. Execute the command randBin(20, .5, 10)/20 as follows:

Press MATH, choose PRB, choose 7:randBin(. Complete command and press ENTER.
(a) Plot a histogram of the 10 values of 𝑝̂ . Set Window parameter to X[-0.05, 1.05]
0.1
and
Y[-2,6]
1
and then TRACE. Is the center of the histogram close to 0.5? Do this several times to see if you get similar results each time.
(a) A histogram is shown below. The center of the histogram is close to 0.5, but there is considerable variation in the overall shape of the histograms for different simulations with only 10 repetitions.

(b) Increase the number of repetitions to 100. The command should read randBin(20, 0.5,
100)/20. Execute the command and then plot a histogram using these 100 values. Don’t change the XMIN and XMAX values, but do adjust the Y-values to Y[-20, 50]
10
to accommodate the taller bars. Is the center close to 0.5? Describe the shape of the distribution.
A histogram for 100 repetitions is shown below (on the left). The distribution is approximately Normal with a center at 0.50.
(c) Define Plot 2 to be a boxplot using L
1
/List1, and use TRACE again. How close is the median (in the boxplot) to the mean (balance point) of the histogram?
(c) The mean and the median are extremely close to one another. See the plot above (on the right).
(d) Note that we didn’t increase the sample size, one the number of repetitions. Did the spread of the distribution change? What would you change to decrease the spread of the distribution?
(d) The spread of the distribution will not change. To decrease the spread, the sample size should be increased.

Chapter 9.1 B Homework
Done on Time:____/5 Completed:____/5
Name_____________________
1.
Should the math department get new batteries?
I have tested a sample of our batteries for life expectancy. Here is what I found. If the mean life expectancy is 17 hours as they claim, then we are happy with the ones we have. The 17 hours of life expectancy is better than what is claimed by the leading competitors. If however it is really less than that, we may be open to buying batteries from a different company.
16.91
17.65
15.80
16.33
16.96
18.83
16.63
15.93
16.22
17.58
16.84
15.81
16.59
15.84
15.63
17.45
17.13
17.42
16.37
16.85
17.10
16.52
16.40
17.04
17.35
17.07
16.37
15.73
15.98
16.74
Do you believe that the companies claim is accurate? Why or why not. Back your argument up with the concepts we learned in class today. Ultimately, is it in our best interest to change batteries in the math department?
The mean time of this sample is 16.7 hours and the standard deviation is .71 hours. I am not sure what the population standard deviation is, but based on the sample that I have, there is no reason to doubt the claim that the mean life expectancy of their batteries is 17 hours. Also, 16.7 is not enough below 17 hours to have any practical significance. If we look at the histogram and the box plat we can see that the data is fairly symmetrical but it does have one outlier. Some batteries are lasting longer than
17 hours, it looks like about 25% of the sample. I would not recommend changing battery brands.

2.
Here are the survival times of 72 guinea pigs from the medical experiment described in Example
2.14 (page 151 in your book). Consider these 72 animals to be the population of interest.
43
80
45
80
53
81
56
81
56
81
57
82
58
83
66
84
67
88
73
89
74
91
79
83
91 92 92 97 99 99 100 100 101 102 102 102
103 104 107 108 109 113 114 118 121 123 126 128
137 138 139 144 145 147 156 162 174 178 179 184
191 198 211 214 243 249 329 380 403 511 522 598
(a) Make a histogram of the 72 survival times. This is the population distribution. It is strongly skewed right.
(b) Find the mean of the 72 survival times. This is the population 𝜇 . Mark 𝜇 on the x axis of your histogram.

(c) Label the members of the population 01 to 72 and use Table B to choose an SRS of size n =
12. What is the mean survival time 𝑥̅ for your sample? Mark the value of 𝑥̅ with a point on the axis of your histogram from (a)
(d) Choose four more SRSs of size 12, using different parts of table B. Find 𝑥̅ for each sample and mark the values on the axis of your histogram from (a). Would you be surprised if all fixe 𝑥̅ ’s fell on the same side of 𝜇 ? Why?
(e) If you chose all possible SRSs of size 12 from this population and made a histogram of the 𝑥̅ values, where would you expect the center of this sampling distribution to lie?
(b) 141.847μ�days.
(c) Means will vary with samples. The mean of our first sample was 120.833.
(d) Our four additional samples produced means of 183.4, 212.8, 127.3, and 119.7. It would be unlikely
(though not impossible) for all five values to fall on the same side ofμ. This is one implication of the unbiasedness ofx: Some values will be higher and some lower thanμ, but not necessarily a 50/50 split.
(e) The mean of the (theoretical) sampling distribution is141.847μ�.
3.
In Internal Revenue Service plans to examine an SRS of individual federal income tax returns from each state. One variable of interest is the proportion of returns claiming itemized deductions. The total number of tax returns in each state varies from over 15 million in
California to about 240,000 in Wyoming.
(a) Will the sampling variability of the sample proportion change from state to state if an SRS of
2000 tax returns is selected in each state? Explain your answer.
(b) Will the sampling variability of the sample proportion change from state to state if an SRS of
1% of all tax returns is selected in each state? Explain your answer.
(a) Since the smallest number of total tax returns (i.e., the smallest population) is still more than
100 times the sample size, the variability will be (approximately) the same for all states. (b) Yes, it will change—the sample taken from Wyoming will be about the same size, but the sample in, e.g., California will be considerably larger, and therefore the variability will be smaller.

4.
Label each sampling distribution below as having large or small bias and as having large or small variability.
(a) Large bias, large variability. (b) Small bias, small variability. (c) Small bias, large variability. (d)
Large bias, small variability.