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# Hypothesis Tests

1. The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean µ = 74 and a variance &#417; ² = 8. Would you still consider &#417; ² = 8 to be a valid value of the variance if a random sample of 20 students who take this placement test this year obtain a value of s ² = 20?

2. A manufacturing firm claims that the batteries used in their electronic games will last an average of 30 hours. To maintain this average, 16 batteries are tested each month. If the computed t-value falls between -t 0.025 and t 0.025 , the firm is satisfied with its claim. What conclusion should the firm draw from a sample that has a mean x = 27.5 hours and a standard deviation s = 5 hours? Assume the distribution of battery lives to be approximately normal.

3. A maker of a certain brand of low-fat cereal bars claims that their average saturated fat content is 0.5 gram. In a random sample of 8 cereal bars of this brand the saturated fat content was 0.6, 0.7, 0.7, 0.3, 0.4, 0.5, 0.4, and 0.2. Would you agree with the claim? Assume a normal distribution.

4. Two independent experiments are being run in which two different types of paints are compared. Eighteen specimens are painted using type A and the drying time, in hours, is recorded on each. The same is done with type B. The population standard deviations are both known to be 1.0. Assuming that the mean drying time is equal for the two types of paint, find P(XA - XB > 1.0), where XA and XB are average drying times for samples of
size n = n = 18.

a) Does this seem to be a reasonable result if the two population mean drying times truly are equal?

b) If someone did the experiment 10,000 times under the condition that µA = µB, in how many of those 10,000 experiments would there be a difference xA - xB that is as large (or larger) as 1.0?

See attached file for full problem description.

#### Solution Preview

1. The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean µ = 74 and a variance &#417; ² = 8. Would you still consider &#417; ² = 8 to be a valid value of the variance if a random sample of 20 students who take this placement test this year obtain a value of s ² = 20?

To do this, we need to do an F test. You simply take the ratio of the two variances and compare that number to an F-distribution.

F = 20/8 = 2.5

Compared to an F distribution with 19 and infinite degrees of freedom (for the numerator and denominator, respectively), you can reject the null hypothesis that the sample came from a population with a variance of 20 at the 0.05 level.

You should not consider 8 to still be a valid value of the population variance.

2. A manufacturing firm claims that the batteries used in their electronic games will last an average of 30 hours. To maintain this average, 16 batteries are tested each month. If the computed t-value falls between -t 0.025 and t 0.025 , the firm is satisfied with its claim. ...

\$2.19