Chapter 7 Classwork Review

Chapter 7 Classwork Review

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1.The weight of the eggs produced by a certain breed of hen is Normally distributed with mean 65 grams (g) and standard deviation 5 g.

(a) Calculate the probability that a randomly selected egg weighs between 62.5 g and 68.75 g. Show your work.

(b) Think of cartons of such eggs as SRSs of size 12 from the population of all eggs. Calculate the probability that the mean weight of the eggs in a carton falls between 62.5 g and 68.75 g. Show your work.

(c) Did you need to know that the population distribution of egg weights was Normal in order to complete parts A. or B.? Justify your answer.

2.Companies are interested in the demographics of those who listen to the radio programs they sponsor. A radio station has determined that only 20% of listeners phoning in to a morning talk program are male. The station management wonders if adding a male host to the program will increase the proportion of callers who are male. After adding the male host, the station records the gender of 200 people who phone in to the program during a particular week. The station is willing to view these 200 callers as an SRS from the population of all those who call in to this program.

(a) For the moment, assume that the addition of the male host has no effect on the proportion of callers who are male. If is the proportion of callers in the sample who are male, what are the mean and standard deviation of the sampling distribution of ?

(b) What assumption are you making when you use the formula for the standard deviation of in this setting?

(c)In fact, during this particular week, 50 of the 200 callers were male. Does this provide sufficient evidence to suggest that the proportion of male callers has increased from 20%? Support your answer with an appropriate probability calculation. [Assume all necessary conditions have been met.]

3.Buying a year’s worth of textbooks for college can be expensive! Consider a large population of college students for whom the distribution of the annual cost of textbooks is slightly skewed to the left. Here is the five-number summary for the annual cost of textbooks for this population:

Minimum = 80 Quartile 1 = 215Median = 335Quartile 3 = 380Maximum = 440

Suppose we take random samples of size 32 from this population and calculate the interquartile range (IQR) for each of our samples. Below is a dotplot of the IQR from 50 such samples.

(a) Briefly explain what the dot at 240 represents.

(b) Is the sample IQR is an unbiased estimator of the population IQR? Justify your answer.

4. A certain beverage company is suspected of underfilling its cans of soft drink. The company advertises that its cans contain, on average, 12 ounces of soda with standard deviation 0.4 ounce. For the questions that follow, suppose that the company is telling the truth.

(a) Can you calculate the probability that a single randomly selected can contains 11.9 ounces or less? If so, do it. If not, explain why you cannot.

(b) A quality control inspector measures the contents of an SRS of 50 cans of the company’s soda and calculates the sample mean . What are the mean and standard deviation of the sampling distribution of for samples of size n = 50?

(c) The inspector in part B. obtains a sample mean of ounces. Calculate the probability that a random sample of 50 cans produces a sample mean amount of 11.9 ounces or less. Be sure to explain why you can use a Normal calculation.

(d) What would you conclude about whether the company is underfilling its cans of soda? Justify your answer.

5.A small internet mail-order company keeps track of the number of orders it fills per day for many years and determines that the distribution of the variable “orders filled per day” is somewhat right-skewed and has the following five-number summary:

Minimum = 20 / Quartile 1 = 32 / Median = 46 / Quartile 3 = 63 / Maximum = 80

Suppose we take random samples of size 40 from this distribution and calculate the range for each of our samples. Below is a dotplot of the ranges from 50 such samples.

Is the sample range an unbiased estimator of the population range? Use the dotplot above to justify your answer.

6.An opinion poll asks a sample of 500 adults (an SRS) whether they favor giving parents of school-age children vouchers that can be exchanged for education at any public or private school of their choice. Each school would be paid by the government on the basis of how many vouchers it collected. Suppose that in fact 45% of the population favor this idea.

(a) What is the mean of the sampling distribution of , the proportion of adults in samples of 500 who favor giving parents of school-age children these vouchers?

(b) What is the standard deviation of ?

(d) What is the probability that more than half of the sample are in favor? Show your work.

Chapter 7 Classwork Review

Answer Section

OTHER

1.ANS:

B. C. Yes. The calculations in both A. and B. assumed the Normality of the underlying distribution. In A. the population was given as Normal. In B., we would not be able to assume the Normality of the sampling distribution because the sample size is less than 30.

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2.ANS:

A. B. We can use this formula if we assume that there are more than 10(200)=2000 listeners who call in to the program.

C. The probability of getting 50 or more males in 200 callers if the true proportion of males is still 0.20 is . Roughly 1 out of 25 times, we will get this many or more male callers. This is probably unusual enough to suggest that the true proportion of male listeners is higher than 0.20.

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3.ANS:

A. The dot at 240 represents the interquartile range of one of the 50 samples taken from this population. B. Responses may vary. If the response judges the mean of the sample IQRs to be about 165, then “unbiased” is the correct conclusion. If the response judges that the mean is higher (it is actually 171.1), then “biased” is the correct conclusion.

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4.ANS:

A. No. We don’t know the shape of the distribution, so we can’t calculate this probability.

B. and

C. Since n = 50, which is greater than 30, we can use the Normal probability distribution.

D. If the true mean amount of soda in the cans is 12 ounces, there is about a 4% chance of getting a sample mean as low or lower than 11.9 ounces. This result is unlikely enough to make us suspicious and lead us to conclude that the company is under-filling its cans of soda!

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5.ANS:

Sample range is not an unbiased estimator of population range. The population range is 80 – 20 = 60. The range of a sample will only be this large if the population’s minimum and maximum values in the distribution are both in the sample. Otherwise, the sample range will be smaller. Thus the mean of the sampling distribution of sample ranges will be somewhere below 60. In this particular case, the mean appears to be closer to 57.

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6.ANS:

A. B.. C. and . So the sampling distribution is approximately Normal.

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