Chapter 7
1. A sample size of 1000 is large enough to conclude that the sampling distribution of p is a normal distribution, when the estimate of the population proportion is .996.
2. The standard deviation of all possible sample proportions decreases as the sample size decreases. FALSE It increases when n gets smaller.
3. If the population is normally distributed then the sample mean is normally distributed for any sample size.
4. The reason sample variance has a divisor of n-1 rather than n is that it makes the standard deviation an unbiased estimator of the population standard deviation.
5. The mean of the sampling distribution of is always equal to the mean of the sampled population.
6. First a confidence interval is constructed without using the finite population correction factor. Then, for the same identical data, a confidence interval is constructed using the finite population correction factor. The width of the interval with the finite population correction factor is wider than the confidence interval without the finite population correction factor.
False. Look at the formula for finite population correction. It reduces the standard deviation.
7. When the population is normally distributed and the population standard deviation is unknown, then for any sample size n, the sampling distribution of X ¯ is based on the t distribution.
TRUE When we substitute the estimated standard deviation in the formula, the distribution becomes t-distribution instead of Z distribution.
8. When the level of confidence and sample standard deviation remain the same, a confidence interval for a population mean based on a sample of n=200 will be wider than a confidence interval for a population mean based on a sample of n= 150. False
9. When the level of confidence and the sample size remain the same, a confidence interval for a population mean µ will be narrower, when the sample standard deviation s is small than when s is large. True
10. When the level of confidence and sample proportion p remain the same, a confidence interval for a population proportion p based on a sample of n=100 will be narrower than a confidence interval for p based on a sample of n=400. FALSE
11. The sample mean, the sample proportion and the sample standard deviation are all unbiased estimators of the corresponding population parameters. FALSE The Sample Standard Deviation is not an unbiased estimator
12. Assuming the same level of significance , as the sample size increases, the value of t/2 approaches the value of z/2. TRUE
Multiple Choices
Chapter 7
1. A manufacturing company measures the weight of boxes before shipping them to the customers. If the box weights have a population mean and standard deviation of 90 lbs. and 24 lbs. respectively, then based on a sample size of 36 boxes, the probability that the average weight of the boxes will exceed 94 lbs. is:
A. 15.87%
B. 84.13%
C. 34.13%
D. 56.36%
E. 16.87%
The σ_(X ¯ )= 24/√36 = 4. Therefore the Z score is (94-90)/4 = 1.0
P(Z ≥ 1.0) = 1 - 0.8413 = 0.1587 or 15.87%.
2. The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent more than 90 days overdue. The historical records of the company show that over the past 8 years 13 percent of the accounts are delinquent. For this quarter, the auditing staff randomly selected 250 customer accounts. What is the probability that more than 40 accounts will be classified as delinquent?
A. 42.07%
B. 92.07%
C. 7.93%
D. 40.15%
E. 90.15%
Here π = 0.13, n = 250, and 1- π = 0.87. We have nπ = 32.5 and n(1- π) = 217.5, both greater than 10, and nπ (1- π) = 28.275 > 10. So, normal approximation without continuity correction is appropriate. The standard error of p = σp = √(0.13*0.87/250) = 0.0213
Next, 40 accounts out of 250 in proportion is 40/250 = 0.16. The question refers to more than 40. Therefore, the question is P(p ≥ 0.16)?
Using the standardization process with µp = 0.13 (the population proportion) and σp = 0.0213, we have P(Z ≥ (0.16-0.13)/(0.0213) ) = P(Z ≥ 1.41) = 1- P(Z ≤ 1.41) = 1- 0.9207 = 0.0793 from the table or MegaStat.
3. If we have a sample size of 100 and the estimate of the population proportion is 0.10, the mean of the sampling distribution of the sample proportion is:
A. 0.009
B. 0.10
C. 0.03
D. 0.90
E. 0.09
4. Consider a sampling distribution formed based on n = 10. The standard deviation of the population of all sample means σ_(X ¯ )is ______________ more than the standard deviation of the population of individual measurements:
A. Always
B. Sometimes
C. Never
Should be obvious from the formula for the standard deviation σ_(X ¯ ).
5. A manufacturing company measures the weight of boxes before shipping them to the customers. If the box weights have a population mean and standard deviation of 90 lbs. and 24 lbs. respectively, then based on a sample size of 36 boxes, the probability that the average weight of the boxes will be be less than 84 lbs. is:
A. 16.87%
B. 93.32%
C. 43.32%
D. 6.68%
E. 84.13%
The σ_(X ¯ )= 24/√36 = 4. Therefore the Z score is (84-90)/4 = -1.5
P(Z ≤ -1.5) = 0.0.0668
6. If a population distribution is known to be normal, then it follows that:
A. The sample mean must equal the population mean
B. The sample mean must equal the population mean for large samples
C. The sample standard deviation must equal the population standard deviation
D. All of the above
E. None of the above
7. According to a hospital administrator, historical records over the past 10 years have shown that 20% of the major surgery patients are dissatisfied with after-surgery care in the hospital. A scientific poll based on 400 hospital patients has just been conducted.
Sixty-four (64) patients indicated that they were dissatisfied with the after surgery care. What are the mean and the standard deviation of the sampling distribution of p ?
A. 16% and .034%
B. 20% and 1.83%
C. 20% and 2%
D. 20% and .034%
E. 20% and 16%
Chapter 8
8. The width of a confidence interval will be:
A. Narrower for 95% confidence than 99% confidence
B. Wider for a sample size of 100 than for a sample size of 50
C. Wider for 90% confidence than 95% confidence
D. Wider when the sample standard deviation (s) is small than when s is large
9. When the level of confidence and sample size remain the same, a confidence interval for a population proportion p will be __________ when p(1-p) is larger than when p(1-p) is smaller.
A. Narrower
B. Wider
C. Neither A nor B, they will be the same
D. Cannot tell from the information given
10. A confidence interval increases in width as
A. The level of confidence increases
B. n decreases
C. s increases
D. All of the above
E. None of the above
11. In a manufacturing process a random sample of 9 bolts manufactured has a mean length of 3 inches with a standard deviation of .3 inches. What is the 95% confidence interval for the true mean length of the bolt?
A. 2.804 to 3.196