Normal Distributions and Sampling Distributions

Lauren is enrolled in a very large college calculus class. On the first exam, the class mean was 75 and the standard deviation was 10. On the second exam, the class mean was 70 and the standard deviation was 15. Lauren scored 85 on both exams. Assuming the scores on each exam were approximately normally distributed, on which exam did Lauren score better relative to the rest of the class?

• A) She scored much better on the first exam.

• B) She scored much better on the second exam.

• C) She scored about equally well on both exams.

• D) It is impossible to tell because the class size is not given.

• E) It is impossible to tell because the correlation between the two sets of exam scores is not

  given.

Let X represent a random variable whose distribution is normal, with a mean of 100 and a standard deviation of 10. Which of the following is equivalent to P(X > 115)?

• A) P(X < 115)

• B) P(X ≤ 115)

• C) P(X < 85)

• D) P(85 < X < 115)

• E) 1 - P(X < 85)

The heights of a population of adult male red kangaroos are approximately normally distributed with a mean height of 1.20 meters and a standard deviation of 0.10 meters. The heights of a population of adult male grey kangaroos are approximately normally distributed with a mean height of 1.75 meters and a standard deviation of 0.25 meters. A certain adult male grey kangaroo is 2.15 meters tall. This kangaroo would have the same standardized height (z-score) as an adult male red kangaroo whose height, in meters, is equal to which of the following?

• A. 1.08

• B. 1.20

• C. 1.30

• D. 1.36

• E. 1.45

• A) About 64 children in the sample have heights that are greater than 49 inches.

• B) About 200 children in the sample have heights that are between 45 and 49 inches.

• C) About 200 children in the sample have heights that are greater than 47 inches.

• D) About 336 children in the sample have heights that are greater than 45 inches.

• E) About 380 children in the sample have heights that are between 43 and 51 inches

Suppose the distribution of weights of adult dogs of a particular breed is strongly skewed right with a mean of 15 pounds and a standard deviation of 4 pounds. Describe the sampling distribution of sample means for a random sample of 40 dogs from the population.

• A) The sampling distribution will be strongly skewed right with a mean of 15 pounds and a standard deviation of 4 pounds.

• B) The sampling distribution will be strongly skewed right with a mean of 15 pounds and a standard deviation of 0.632 pounds.

• C) The sampling distribution will be approximately normally distributed with a mean of 15 pounds and standard deviation of 4 pounds.

• D) The sampling distribution will be approximately normally distributed with a mean of 15 pounds and standard deviation of 0.1 pounds.

• E) The sampling distribution will be approximately normally distributed with a mean of 15 pounds and standard deviation of 0.632 pounds.

Forty-five percent of households in a large metropolitan area have at least one pet. What is the approximate probability of obtaining a random sample of 100 households in the area, in which 35 or less have at least one pet?

• A) 0.011

• B) 0.018

• C) 0.022

• D) 0.978

• E) 0.982

The distribution of the diameters of a particular variety of cantaloupe is approximately normal with a standard deviation of 0.7 inches. If the diameter of a cantaloupe is at the 73rd percentile, how does this diameter compare to the mean?

• A) 0.511 inches below the mean

• B) 0.429 inches below the mean

• C) 0.511 inches above the mean

• D) 0.429 inches above the mean

• E) 0.610 inches above the mean

The weights of adult male Chinook salmon caught in the rivers along the western coast of Washington are normally distributed with a mean of 25 kilograms and standard deviation 8 kilograms. Suppose regulations require that only 3 salmon can be caught per day by any one fisherman. Assuming that on his next trip Jim catches three salmon, what is the approximate probability that the average weight of the fish on his stringer is at least 30 kilograms?

• A) 0.03

• B) 0.14

• C) 0.27

• D) 0.41

• E) 0.45

Suppose that public opinion in a large city is 72 percent in favor of allowing a large recycling plant to be constructed using taxpayer funds and 28 percent against such a build. If a random sample of 300 people from this city is interviewed, what is the approximate probability that more than 100 of these people will be against increasing taxes?

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The heights of mature maple trees are approximately normally distributed with a mean of 80 feet and standard deviation 12.5 feet. What proportion of mature maple trees are between 65 and 90 feet tall?

• A) 0.1151

• B) 0.2119

• C) 0.6731

• D) 0.7881

• E) 0.8849