Sanderson M. Smith

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FOUR HYPOTHESIS TESTING SITUATIONS

A coin is flipped 500 times and results in an outcome of 262 heads and 238 tails. At the 5% level of significance, test the hypothesis that the coin is fair.

Analysis: Using proportions, we have p(hat) = 262/500 = 0.524. This is a two-tail test.

H_o: p = 0.5.

H_a: p is not equal to 0.5.

If H_o is true, then

m_{p(hat )}= 0.5.

s_p(hat) = SQRT[(.5)(.5)/500] = 0.02236.

z_.524 = (0.524-0.5)/.02236 = 1.07.

Critical values at the 5% level of significance (2 tail) are z < -1.96 and z > 1.96. Our calculated sample z score is not in the critical region. Hence, we fail to reject H_o at this level of significance. There is not strong evidence to suggest that the coin is not a fair coin.

An advertisement says that a manufactured product weight at least 36 ounces. Production standards are such that the allowed standard deviation is 0.5 ounces. A random sample of 75 items is selected and found to have a mean weight of 35.81 ounces. Test, at the 5% level of significance, that the production standards are being met.

Analysis: This is a two-tail test.

H_o: m = 36.

H_a: m is not equal to 36.

If H_o is true, the Central Limit Theorem states that the means of all samples of size 75 is approximately normally distributed with mean = 36 and standard deviation = 0.5/SQRT(75) = 0.0577.

z_35.81 = (35.81-36)/0.0577 = -3.29.

Critical values at the 5% level of significance (2 tail) are z < -1.96 and z > 1.96. Our calculated value of z is in the critical region. Hence, we reject Ho at the 5% level of significance. Based on the sample, there is strong evidence to suggest that production standards are not being met.

Centerville has a population of 185,234 registered voters. A randomly selected sample of 1,000 registered voters were asked if they would support the creation of a school bond to finance a new classroom building at a local high school. Here are the results:

Test, at the 1% level of significance, that at least 50% of the registered voters in Centerville will not support the creation of the bond.

Analysis: This is a one-tail test.

H_o: p = 0.5

H_a: p < 0.5

If H_o is true, then

m_{p(hat )}= 0.5.

s_p(hat) = SQRT[(.5)(.5)/1000] = 0.0158.

z_.48 = (0.475-0.5)/.0158 = -1.58.

Critical values at the 1% level (one-tail) are z < -2.33. Our calculated value of z is not in the critical region. Hence, we fail to reject H_o at the 1% level of significance. There is not strong evidence to suggest that the bond measure will fail.

In a mass production situation, a spherical object should have a diameter of 0.008 inches with an allowable standard deviation of 0.0001 inches. A random sample of 50 items yields a mean diameter of 0.008009 inches. Test, at the 5% of significance, that the production process is yielding diameters more than the specified 0.008 inches.

Analysis: This is a one-tail test.

H_o: m = 0.008

H_a: m > 0.008

If H_o is true, then the Central Limit Theorem states that the means of all samples of size 50 will be approximately normally distributed with mean = 0.008 and standard deviation = 0.0001/SQRT(50) = 0.000014.

z_0.008009 = (.008009-0.008)/0.000014 = 0.64.

At the 5% level of significance (one-tail), the critical values of z are z > 1.65. Our calculated sample z score is not in the critical region. Hence, we fail to reject H_o at the 5% level of significance. There is not strong evidence to suggest that the production process is yielding diameters that are greater than 0.008 inches.

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