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6.E: Sampling Distributions (Exercises)

These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here. In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.

6.1: The Mean and Standard Deviation of the Sample Mean

Q6.1.1

Random samples of size 225 are drawn from a population with mean 100 and standard deviation 20. Find the mean and standard deviation of the sample mean.

Q6.1.2

Random samples of size 64 are drawn from a population with mean 32 and standard deviation 5. Find the mean and standard deviation of the sample mean.

S6.1.2

A population has mean 75 and standard deviation 12.

Q6.1.3

Random samples of size 121 are taken. Find the mean and standard deviation of the sample mean.

Q6.1.4

How would the answers to part (a) change if the size of the samples were 400 instead of 121?

 

A population has mean 5.75 and standard deviation 1.02.

Random samples of size 81 are taken. Find the mean and standard deviation of the sample mean.

How would the answers to part (a) change if the size of the samples were 25 instead of 81?

 

Answers

μX−−=100σX−−=1.33

μX−−=75σX−−=1.09

μX−− stays the same but σX−− decreases to 0.6

6.3: The Sample Proportion

Exercises

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Basic

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  1. The proportion of a population with a characteristic of interest is p = 0.37. Find the mean and standard deviation of the sample proportion Pˆ obtained from random samples of size 1,600.

  2. The proportion of a population with a characteristic of interest is p = 0.82. Find the mean and standard deviation of the sample proportion Pˆ obtained from random samples of size 900.

  3. The proportion of a population with a characteristic of interest is p = 0.76. Find the mean and standard deviation of the sample proportion Pˆ obtained from random samples of size 1,200.

  4. The proportion of a population with a characteristic of interest is p = 0.37. Find the mean and standard deviation of the sample proportion Pˆ obtained from random samples of size 125.

  5. Random samples of size 225 are drawn from a population in which the proportion with the characteristic of interest is 0.25. Decide whether or not the sample size is large enough to assume that the sample proportion Pˆ is normally distributed.

  6. Random samples of size 1,600 are drawn from a population in which the proportion with the characteristic of interest is 0.05. Decide whether or not the sample size is large enough to assume that the sample proportion Pˆ is normally distributed.

  7. Random samples of size n produced sample proportions pˆ as shown. In each case decide whether or not the sample size is large enough to assume that the sample proportion Pˆ is normally distributed.

    1. n = 50, pˆ=0.48
    2. n = 50, pˆ=0.12
    3. n = 100, pˆ=0.12
  8. Samples of size n produced sample proportions pˆ as shown. In each case decide whether or not the sample size is large enough to assume that the sample proportion Pˆ is normally distributed.

    1. n = 30, pˆ=0.72
    2. n = 30, pˆ=0.84
    3. n = 75, pˆ=0.84
  9. A random sample of size 121 is taken from a population in which the proportion with the characteristic of interest is p = 0.47. Find the indicated probabilities.

    1. P(0.45Pˆ0.50)
    2. P(Pˆ0.50)
  10. A random sample of size 225 is taken from a population in which the proportion with the characteristic of interest is p = 0.34. Find the indicated probabilities.

    1. P(0.25Pˆ0.40)
    2. P(Pˆ0.35)
  11. A random sample of size 900 is taken from a population in which the proportion with the characteristic of interest is p = 0.62. Find the indicated probabilities.

    1. P(0.60Pˆ0.64)
    2. P(0.57Pˆ0.67)
  12. A random sample of size 1,100 is taken from a population in which the proportion with the characteristic of interest is p = 0.28. Find the indicated probabilities.

    1. P(0.27Pˆ0.29)
    2. P(0.23Pˆ0.33)

Applications

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  1. Suppose that 8% of all males suffer some form of color blindness. Find the probability that in a random sample of 250 men at least 10% will suffer some form of color blindness. First verify that the sample is sufficiently large to use the normal distribution.

  2. Suppose that 29% of all residents of a community favor annexation by a nearby municipality. Find the probability that in a random sample of 50 residents at least 35% will favor annexation. First verify that the sample is sufficiently large to use the normal distribution.

  3. Suppose that 2% of all cell phone connections by a certain provider are dropped. Find the probability that in a random sample of 1,500 calls at most 40 will be dropped. First verify that the sample is sufficiently large to use the normal distribution.

  4. Suppose that in 20% of all traffic accidents involving an injury, driver distraction in some form (for example, changing a radio station or texting) is a factor. Find the probability that in a random sample of 275 such accidents between 15% and 25% involve driver distraction in some form. First verify that the sample is sufficiently large to use the normal distribution.

  5. An airline claims that 72% of all its flights to a certain region arrive on time. In a random sample of 30 recent arrivals, 19 were on time. You may assume that the normal distribution applies.

    1. Compute the sample proportion.
    2. Assuming the airline’s claim is true, find the probability of a sample of size 30 producing a sample proportion so low as was observed in this sample.
  6. A humane society reports that 19% of all pet dogs were adopted from an animal shelter. Assuming the truth of this assertion, find the probability that in a random sample of 80 pet dogs, between 15% and 20% were adopted from a shelter. You may assume that the normal distribution applies.

  7. In one study it was found that 86% of all homes have a functional smoke detector. Suppose this proportion is valid for all homes. Find the probability that in a random sample of 600 homes, between 80% and 90% will have a functional smoke detector. You may assume that the normal distribution applies.

  8. A state insurance commission estimates that 13% of all motorists in its state are uninsured. Suppose this proportion is valid. Find the probability that in a random sample of 50 motorists, at least 5 will be uninsured. You may assume that the normal distribution applies.

  9. An outside financial auditor has observed that about 4% of all documents he examines contain an error of some sort. Assuming this proportion to be accurate, find the probability that a random sample of 700 documents will contain at least 30 with some sort of error. You may assume that the normal distribution applies.

  10. Suppose 7% of all households have no home telephone but depend completely on cell phones. Find the probability that in a random sample of 450 households, between 25 and 35 will have no home telephone. You may assume that the normal distribution applies.

Additional Exercises

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  1. Some countries allow individual packages of prepackaged goods to weigh less than what is stated on the package, subject to certain conditions, such as the average of all packages being the stated weight or greater. Suppose that one requirement is that at most 4% of all packages marked 500 grams can weigh less than 490 grams. Assuming that a product actually meets this requirement, find the probability that in a random sample of 150 such packages the proportion weighing less than 490 grams is at least 3%. You may assume that the normal distribution applies.

  2. An economist wishes to investigate whether people are keeping cars longer now than in the past. He knows that five years ago, 38% of all passenger vehicles in operation were at least ten years old. He commissions a study in which 325 automobiles are randomly sampled. Of them, 132 are ten years old or older.

    1. Find the sample proportion.
    2. Find the probability that, when a sample of size 325 is drawn from a population in which the true proportion is 0.38, the sample proportion will be as large as the value you computed in part (a). You may assume that the normal distribution applies.
    3. Give an interpretation of the result in part (b). Is there strong evidence that people are keeping their cars longer than was the case five years ago?
  3. A state public health department wishes to investigate the effectiveness of a campaign against smoking. Historically 22% of all adults in the state regularly smoked cigars or cigarettes. In a survey commissioned by the public health department, 279 of 1,500 randomly selected adults stated that they smoke regularly.

    1. Find the sample proportion.
    2. Find the probability that, when a sample of size 1,500 is drawn from a population in which the true proportion is 0.22, the sample proportion will be no larger than the value you computed in part (a). You may assume that the normal distribution applies.
    3. Give an interpretation of the result in part (b). How strong is the evidence that the campaign to reduce smoking has been effective?
  4. In an effort to reduce the population of unwanted cats and dogs, a group of veterinarians set up a low-cost spay/neuter clinic. At the inception of the clinic a survey of pet owners indicated that 78% of all pet dogs and cats in the community were spayed or neutered. After the low-cost clinic had been in operation for three years, that figure had risen to 86%.

    1. What information is missing that you would need to compute the probability that a sample drawn from a population in which the proportion is 78% (corresponding to the assumption that the low-cost clinic had had no effect) is as high as 86%?
    2. Knowing that the size of the original sample three years ago was 150 and that the size of the recent sample was 125, compute the probability mentioned in part (a). You may assume that the normal distribution applies.
    3. Give an interpretation of the result in part (b). How strong is the evidence that the presence of the low-cost clinic has increased the proportion of pet dogs and cats that have been spayed or neutered?
  5. An ordinary die is “fair” or “balanced” if each face has an equal chance of landing on top when the die is rolled. Thus the proportion of times a three is observed in a large number of tosses is expected to be close to 1/6 or 0.16. Suppose a die is rolled 240 times and shows three on top 36 times, for a sample proportion of 0.15.

    1. Find the probability that a fair die would produce a proportion of 0.15 or less. You may assume that the normal distribution applies.
    2. Give an interpretation of the result in part (b). How strong is the evidence that the die is not fair?
    3. Suppose the sample proportion 0.15 came from rolling the die 2,400 times instead of only 240 times. Rework part (a) under these circumstances.
    4. Give an interpretation of the result in part (c). How strong is the evidence that the die is not fair?

Answers

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  1. μPˆ=0.37σPˆ=0.012

  2.  
  3. μPˆ=0.76σPˆ=0.012

  4.  
  5. p±3pqn=0.25±0.087, yes

  6.  
    1. pˆ±3pˆqˆn=0.48±0.21, yes
    2. pˆ±3pˆqˆn=0.12±0.14, no
    3. pˆ±3pˆqˆn=0.12±0.10, yes
  7.  
    1. 0.4154
    2. 0.2546
  8.  
    1. 0.7850
    2. 0.9980
  9.  
  1. p±3pqn=0.08±0.05

    and

    [0.03,0.13][0,1],0.1210

  2.  
  3. p±3pqn=0.02±0.01

    and

    [0.01,0.03][0,1],0.9671

  4.  
    1. 0.63
    2. 0.1446
  5.  
  6. 0.9977

  7.  
  8. 0.3483

  9.  
  1. 0.7357

  2.  
    1. 0.186
    2. 0.0007
    3. In a population in which the true proportion is 22% the chance that a random sample of size 1500 would produce a sample proportion of 18.6% or less is only 7/100 of 1%. This is strong evidence that currently a smaller proportion than 22% smoke.
  3.  
    1. 0.2451
    2. We would expect a sample proportion of 0.15 or less in about 24.5% of all samples of size 240, so this is practically no evidence at all that the die is not fair.
    3. 0.0139
    4. We would expect a sample proportion of 0.15 or less in only about 1.4% of all samples of size 2400, so this is strong evidence that the die is not fair.