Self Test 5 for Probability and Statistics

Sampling Theory

 

This test was constructed by Zauresh Atakhanova based on the Schaum's Outline Theory and Problems of Probability and Statistics by Murray R. Spiegel.  If you need more review refer also to this outline.

 

 

 

Sampling Distribution of Means

 

1. True False A population consists of the four numbers 2,3,6,9.Consider all possible samples of size two which can be drawn with replacement from this population.  Find (a) the mean of the population, (b) the standard deviation of the population, (c) the mean of the sampling distribution of means, (d) the standard deviation of the sampling distribution of means, i.e. the standard error of the means.  

a)      5

b)      5.48

c)      5

d)      2.74

 

2.True False. Assume that the heights of 3500 female students at a university are normally distributed with mean 67.0 inches and standard deviation of 3.0 inches.  If 80 samples consisting of 25 students are obtained, what would be the expected mean and standard deviation of the resulting sampling distribution of means if sampling were done with replacement?

 

67.0 and 0.6 respectively

 

 

3.True False. In how many samples of Problem 5.2 would you expect to find the mean (a) between 65.8 and 67.3 inches, (b) less than 65.4 inches.

 

a)      73

b)      10

 

Sampling Distributions of Differences and Sums

 

4.True False. The electric bulbs of brand A have a mean lifetime of 1500 hours with a standard deviation of 200 hours, while those of brand B have a mean lifetime of 1300 hours with a standard deviation of 100 hours.  If random samples of 125 bulbs of each brand are tested, what is the probability that the brand A bulbs will have a mean lifetime which is at least (a) 160 hours, (b) 250 hours more than the brand B bulbs?

 

a)      0.7972

b)      0.0022

 

 

 

Case Where Population Variance is Unknown

 

5.True False “If random samples of size n are taken from a normally distributed population then the statistic  has Student’s t distribution with n-1 degrees of freedom.”

 

 

Sampling Distributions of Ratios of Variances

 

6.True False. Two samples of sizes 8 and 10 are drawn from two normally distributed populations with variances 20 and 30 respectively.  The probability that the variance of the first sample is more than twice the variance of the second is 0.01<p<0.05