# Standard Deviation Tests

68. Past experience indicates that the time required for high school seniors to complete a standardized test is a normal random variable with a standard deviation of 6 minutes. Test the hypothesis that sigma = 6 against the alternative that sigma < 6 if a random sample of 20 high school seniors has a standard deviations s = 4.51. Use a 0.05 level of significance.

70. Past data indicate that the amount of money contributed by the working residents of a large city to a volunteer rescue squad is a normal random variable with a standard deviation of $1.40. It has been suggested that the contributions to the rescue squad from just the employees of the sanitation department are much more variable. If the contributions of a random sample of 12 employees from the sanitation department had a standard deviation of $1.75, can we conclude at the 0.01 level of significance that the standard deviation of the contributions of all sanitation workers is greater than that of all workers living in this city?

76. Two types of instruments for measuring the amount of sulfur monoxide in the atmosphere are being compared in an air-pollution experiment. It is desired to determine whether the two types of instruments yield measurements having the same variability. The following readings were recorded for the two instruments:

Instrument A Instrument B

0.86 0.87

0.82 0.74

0.75 0.63

0.61 0.55

0.89 0.76

0.64 0.70

0.81 0.69

0.68 0.57

0.65 0.53

Assuming the populations of measurements to be approximately normally distributed, test the hypothesis that sigma_A does not equal sigma_B.

#### Solution Preview

Each of these problems involves comparing standard deviations. This website gives an explanation of how to do this: http://www.tnstate.edu/ganter/BIO%20311%20Ch13%20OddsEnds.html (it has some typos - mixes up left-tailed test and right-tailed test - in the chi-square section).

One sample test: To compare the standard deviation of a normal population to a given number, we use the chi-square test. The test statistic is:

X2 = (n - 1)(s/s0)2

where n is the sample size, s is the sample standard deviation, and s0 is the number you're comparing it to. Then compare this value to an chi-square distribution with the correct degrees of freedom (df = n - 1).

Two sample test: To compare the standard deviations of two normal populations, we use an F test. The F test is very easy. Calculate the test statistic as

F = s12/s22,

where s1 and s2 are your sample standard deviations and s1 is larger than s2 (so F > 1). Then compare this value to an F-distribution with the correct degrees of freedom (df = n1 - 1, n2 - 1).

68. Past experience indicates that the time required for high school seniors to complete a standardized test is a ...

#### Solution Summary

The solution explains how to do one-sample and two-sample tests using sample standard deviations.