Tests of Means and Proportions (Bruning & Kintz, 1968)
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Ho: population mean is less than or equal to 10.
H1: population mean is greater than 10.
t = (14.44-10)/(4.45/5.916) = 4.44/.752 = 5.916.
Note 5.916 = square root of N = 35.
This value is greater than a critical t value of 2.444 (alpha = .01, one-tailed test, df = 34).
Reject Ho.
Ho: P less than or equal to .5
H1: P greater than .5
z = (.633 – .5)/ (square root of .25/60) = .133/.0645 = 2.062.
Note: .25 = .5x.5.
This value is greater than 1.645, the critical value for a one-tailed test, with alpha = .05, so reject Ho.
(The proportion of time the coin came up heads = 38/60 = .63. As shown above, this proportion is greater than .5, where alpha = .05.)
H0: P less than or equal to .95.
H1: P greater than .95.
z = (.97 – .95)/(square root of .0475/500) = .02/.00974 = 2.053.
Note: .047 = .95x.05.
Assuming alpha = .25, the above value is greater than the critical one of .67, so reject Ho.
(Note, the above value also is greater than 1.645, the critical one-tailed value for alpha = .05, so Ho would have been rejected at alpha = .05.)
Ho: population mean is less than or equal to $250.
H1: population mean is greater than $250.
t = ($275.66 – $250.)/($78.11/5) = 25.66/15.622 = 1.6425.
(Note 5 = square root of N = 25.)
The above value does not exceed the critical one of 1.711, alpha = .05, one-tailed test, df = 24, so fail to reject Ho. Although it can’t be concluded that there was a violation of the guideline, the large variability among the patients (indicated by the large standard deviation) suggests a test with a larger sample size (e.g., n = 50) should be done. (Yes,
it was a “close” decision).
Reference
Bruning, J. L., & Kintz, B.L. (1968). Computational handbook of statistics.
Glenville, IL: Scott-Foresman.
