Quiz 6 Sampling distributions
1. Choose the correct word(s) from the list below to correctly complete the following sentence: If the original population from which the samples were drawn is not normally distributed, then the sampling distribution of the mean will be ______for large sample sizes.
a. normal
*b. approximately normal
c. the same as the original population distribution
d. unidentifiable
e. uniform
2. Which of the following statements is false?
*a. the sampling distribution of the mean will have the same standard deviation as the original population from which the samples were drawn
b. the sampling distribution of the mean will have the same mean as the original population from which the samples were drawn
c. the sampling distribution of the mean will be normal if the original population from which the samples were drawn is normally distributed
d. sample data are used as a basis from which to make probability statements about the true (but unknown) value of the population mean or proportion
e. using information from a sample to reach conclusions about the population from which it was drawn is referred to as inferential statistics
3. Consider a large population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. What is the standard deviation of the sample mean?
*a. 3.125
b. 2.500
c. 3.750
d. 5.625
e. 5.000
4. Consider a large population with a mean of 160 and a standard deviation of 20. A random sample of size 64 is taken from this population. What is the standard deviation of the sample mean?
a. 3.125
*b. 2.500
c. 3.750
d. 5.625
e. 5.000
5. Consider a large population with a mean of 160 and a standard deviation of 30. A random sample of size 64 is taken from this population. What is the standard deviation of the sample mean?
a. 3.125
b. 2.500
*c. 3.750
d. 5.625
e. 5.000
6. Consider a large population with a mean of 160 and a standard deviation of 45. A random sample of size 64 is taken from this population. What is the standard deviation of the sample mean?
a. 3.125
b. 2.500
c. 3.750
*d. 5.625
e. 5.000
7. Consider a large population with a mean of 160 and a standard deviation of 40. A random sample of size 64 is taken from this population. What is the standard deviation of the sample mean?
a. 3.125
b. 2.500
c. 3.750
d. 5.625
*e. 5.000
8. A sample of size n is selected at random from a large population. As n increases, which of the following statements is true?
a. the population standard deviation decreases
*b. the standard deviation of the sample mean decreases
c. the population standard deviation increases
d. the standard deviation of the sample mean increases
e. the standard deviation of the sample mean remains unchanged
9. Which of the following statements is correct?
a. If X is normally distributed then the sample mean is skewed to the right
b. If X is normally distributed then the sample mean is normally distributed with the same mean and variance as X.
*c. If X is not normally distributed then the sample mean is approximately normally distributed as long as the sample size is greater than 30
d. If X is not normally distributed then the sample mean is not normally distributed
e. none of the above statements is correct
10. Why is the Central Limit Theorem so important to the study of sampling distributions?
a. It allows us to disregard the size of the population we are sampling from
b. It allows us to disregard the size of the sample selected when the population is not normal
c. It allows us to disregard the shape of the sampling distribution when the size of the population is large
*d. It allows us to estimate the sampling distribution of any population when the sample size is large enough is large
e. None of the above is a correct statement
11. In a given year, the average annual salary of professional South African soccer players was R189,000 with a standard deviation of R20,500. If a sample of 50 players was taken, what is the probability that the sample mean of their salaries was more than R192,000?
*a. 0.1515
b. 0.3669
c. 0.2451
d. 0.2549
e. 0.3485
12. In a given year, the average annual salary of professional South African soccer players was R189,000 with a standard deviation of R20,500. If a sample of 50 players was taken, what is the probability that the sample mean of their salaries was more than R190,000?
a. 0.1515
*b. 0.3669
c. 0.2451
d. 0.2549
e. 0.3485
13. In a given year, the average annual salary of professional South African soccer players was R189,000 with a standard deviation of R20,500. If a sample of 50 players was taken, what is the probability that the sample mean of their salaries was more than R191,000?
a. 0.1515
b. 0.3669
*c. 0.2451
d. 0.2549
e. 0.3485
14. In a certain stats class, the marks obtained by students on a class test followed a normal distribution with a mean of 68% and a standard deviation of 10%. What is the probability that the mean test mark from a sample of 25 students from the class was more than 72%?
*a. 0.0228
b. 0.0668
c. 0.1587
d. 0.3085
e. 0.9332
15. In a certain stats class, the marks obtained by students on a class test followed a normal distribution with a mean of 68% and a standard deviation of 10%. What is the probability that the mean test mark from a sample of 25 students from the class was more than 71%?
a. 0.0228
*b. 0.0668
c. 0.1587
d. 0.3085
e. 0.9332
16. In a certain stats class, the marks obtained by students on a class test followed a normal distribution with a mean of 68% and a standard deviation of 10%. What is the probability that the mean test mark from a sample of 25 students from the class was more than 70%?
a. 0.0228
b. 0.0668
*c. 0.1587
d. 0.3085
e. 0.9332
17. In a certain stats class, the marks obtained by students on a class test followed a normal distribution with a mean of 68% and a standard deviation of 10%. What is the probability that the mean test mark from a sample of 25 students from the class was more than 69%?
a. 0.0228
b. 0.0668
c. 0.1587
*d. 0.3085
e. 0.9332
18. In a certain stats class, the marks obtained by students on a class test followed a normal distribution with a mean of 68% and a standard deviation of 10%. What is the probability that the mean test mark from a sample of 25 students from the class was more than 65%?
a. 0.0228
b. 0.0668
c. 0.1587
d. 0.3085
*e. 0.9332
19. The average daily temperature in Johannesburg during summer follows a normal distribution with a mean of 27 degrees Celsius and a standard deviation of 15 degrees Celsius. What is the probability that a randomly chosen sample of 10 summer days will have an average temperature of less than 28 degrees?
*a. 0.5832
b. 0.4168
c. 0.3372
d. 0.7357
e. 0.2643
20. The average daily temperature in Johannesburg during summer follows a normal distribution with a mean of 27 degrees Celsius and a standard deviation of 15 degrees Celsius. What is the probability that a randomly chosen sample of 10 summer days will have an average temperature of less than 26 degrees?
a. 0.5832
*b. 0.4168
c. 0.3372
d. 0.7357
e. 0.2643
21. The average daily temperature in Johannesburg during summer follows a normal distribution with a mean of 27 degrees Celsius and a standard deviation of 15 degrees Celsius. What is the probability that a randomly chosen sample of 10 summer days will have an average temperature of less than 25 degrees?
a. 0.5832
b. 0.4168
*c. 0.3372
d. 0.7357
e. 0.2643
22. The average daily temperature in Johannesburg during summer follows a normal distribution with a mean of 27 degrees Celsius and a standard deviation of 15 degrees Celsius. What is the probability that a randomly chosen sample of 10 summer days will have an average temperature of less than 30 degrees?
a. 0.5832
b. 0.4168
c. 0.3372
*d. 0.7357
e. 0.2643
23. The average daily temperature in Johannesburg during summer follows a normal distribution with a mean of 27 degrees Celsius and a standard deviation of 15 degrees Celsius. What is the probability that a randomly chosen sample of 10 summer days will have an average temperature of less than 24 degrees?
a. 0.5832
b. 0.4168
c. 0.3372
d. 0.7357
*e. 0.2643
24. Given a large population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations is greater than 78 is equal to:
*a. 0.0668
b. 0.4332
c. 0.9938
d. 0.1915
e. 0.3085
25. Given a large population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations is greater than 70 is equal to:
a. 0.0668
b. 0.4332
*c. 0.9938
d. 0.1915
e. 0.3085
26. Given a large population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations is greater than 76 is equal to:
a. 0.0668
b. 0.4332
c. 0.9938
d. 0.1915
*e. 0.3085
27. A large population has a mean of 60 and a standard deviation of 8. A sample of 50 observations is taken at random from this population. What is the probability that the sample mean will be between 57 and 62?
*a. 0.9576
b. 0.9960
c. 0.2467
d. 0.3520
e. 0.0247
28. At a computer manufacturing company, the size of computer chips is normally distributed with a mean of 1cm and a standard deviation of 0.1cm. A random sample of 12 computer chips is taken. What is the probability that the sample mean will be between 0.99 and 1.01cm?
*a. 0.27
b. 0.50
c. 0.35
d. 0.70
e. 0.13
29. The time until first failure of a brand of ink jet printers is normally distributed with a mean of 1500 hours and a standard deviation of 200 hours. A large company buys four such printers. What is the probability that the mean lifetime of the four printers is more than 1600 hours?
*a. 0.1587
b. 0.3413
c. 0.0668
d. 0.4332
e. 0.3085
30. The time until first failure of a brand of ink jet printers is normally distributed with a mean of 1500 hours and a standard deviation of 200 hours. A large company buys four such printers. What is the probability that the mean lifetime of the four printers is more than 1650 hours?
a. 0.1587
b. 0.3413
*c. 0.0668
d. 0.4332
e. 0.3085
31. The time until first failure of a brand of ink jet printers is normally distributed with a mean of 1500 hours and a standard deviation of 200 hours. A large company buys four such printers. What is the probability that the mean lifetime of the four printers is more than 1550 hours?
a. 0.1587
b. 0.3413
c. 0.0668
d. 0.4332
*e. 0.3085
32. Assume that the time needed by a worker to perform a maintenance operation is normally distributed with a mean of 70 minutes and a standard deviation of 6 minutes. What is the probability that the average time needed by a sample of 5 workers to perform such a maintenance is between 63 minutes and 68 minutes?
a. 0.150
b. 0.479
c. 0.007
*d 0.222
e. 0.348
33. The diameters of Ping-Pong balls manufactured at a large factory are approximately normally distributed, with a mean of 3cm and a standard deviation of 0.4cm. A random sample of 16 Ping-Pong balls was selected. What is the probability that the sample mean diameter of the Ping-Pong balls will be between 2.7 and 3.1cm?
a. 0.49
c. 0.13
*c. 0.84
d. 0.82
e. 0.34
34. The diameters of Ping-Pong balls manufactured at a large factory are approximately normally distributed, with a mean of 3cm and a standard deviation of 0.4cm. A random sample of 16 Ping-Pong balls was selected. What is the probability that the sample mean diameter of the Ping-Pong balls will be between 2.9 and 3.2cm?
a. 0.49
c. 0.13
c. 0.84
*d. 0.82
e. 0.34
35. Certain electric bulbs produced by a company have a mean lifetime of 1000 hours with a standard deviation of 160 hours. The bulbs are packed in boxes of 100. What is the probability that the average lifetime for a randomly selected box exceeds 1020 hours?
a. 0.3944
*b. 0.1056
c. 0.3315
d. 0.1784
e. 0.7891
36. The mean selling price of new apartments over a year in a small town in the Karoo was R 115 000. The population standard deviation was R25 000. A random sample of 100 new apartment sales from the town was taken. What is the probability that the sample mean selling price was between R113 000 and R117 000?
a. 1.1
*b. 0.5762
c. 0.0143
d. 0.1230
e. 0.0268
37. The mean selling price of new apartments over a year in a small town in the Karoo was R 115 000. The population standard deviation was R25 000. A random sample of 100 new apartment sales from the town was taken. What is the probability that the sample mean selling price was more than R110 000?
*a. 0.9772
b. 0.5620
c. 0.0243
d. 0.8230
e. 0.7268
38. A manufacturing company packages peanuts for South African Airways. The average weight of individual packages is 14.0 grams with a standard deviation of 0.6 grams. For a flight of 144 passengers receiving the peanuts, what is the probability that the average weight of peanuts per pack is less than 13.9 grams?