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 twotail 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} = (0.5240.5)/.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 twotail 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} = (35.8136)/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:



Support creation of school bond 


Against creation of school bond 


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 onetail 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} = (0.4750.5)/.0158 = 1.58.
Critical values at the 1% level (onetail) 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 onetail 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} = (.0080090.008)/0.000014 = 0.64.
At the 5% level of significance (onetail), 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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