Question 1178416: Among 157 African-American men ,the mean systolic blood pressure was 146 mm Hg with a
standard deviation of 27. We wish to know if on the basis of these data, we may conclude that the
mean systolic blood pressure for a population of African-American is greater than 140. Use α=0.01
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sample size = 157
sample mean = 146
sample standard deviation = 27
comparison mean = 140
one tailed high side .01 alpha confidence interval has critical z-score of 2.3263.
this means that the probability of getting a z-score greater than that is .01.
z-score formula = (x - m) / s
x is the raw score of the sample.
m is the population mean given for comparison.
s is the standard error = standard deviation / sqrt (sample size) = 27 / sqrt(157) = 2.1548.
z-score formula becomes:
z = (146 - 140) / 2.1548 = 2.7845.
since that's higher than the critical z-score of 2.3263, then the results are considered significant and the claim that the average blood pressure of the population is higher than 140 is accepted as being very likely.
if a t-score analysis was used, then the critical t-score with 156 degrees of freedom would have been 2.3505.
the t-score of the sample would be the same at 2.7845 which would still be significant becuse it was greater than the critical t-score.
note that, as the sample size gets larger, the critical t-score and z-score results converge.
with a sample size of 157, the difference was 2.3263 versus 2.3505; this was not a large difference.
if the sample size was only 15, the critical z-score would have been the same, but the critical t-score would have been 2.6245, a larger difference.
t-score was indicated in this analysis because the standard deviation was from the sample rather than from the population.
because of the large sample size, it didn't make much difference.
the standard error, which is the standard deviation of the distribution of sample means, was calculated in the same way, namely, s = standard deviation divided by square root of sample size.
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