Question 1135733: In a recent year, the Better Business Bureau settled 75% of complaints they received. (Source: USA Today, March 2, 2009) You have been hired by the Bureau to investigate complaints this year involving computer stores. You plan to select a random sample of complaints to estimate the proportion of complaints the Bureau is able to settle. Assume the population proportion of complaints settled for the computer stores is the 0.75, as mentioned above. Suppose your sample size is 229. What is the probability that the sample proportion will be at least 3 percent more than the population proportion?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! p = .75
q = 1 - p = .25
n = 229
s = sqrt(p * q / n) = sqrt(.75 * .25 / 229) = .0286142848.
x = .78
z = (.78 - .75) / .0286142848 = 1.048427394.
p(z > 1.048427394) =.1472208927.
that's the probability that the sample proportion will be at least 3 percent more than the population proportion.
i used the TI-84 Plus to get these numbers.
you can use an online calculator to do the same.
following are the results of using the online calculator at http://davidmlane.com/hyperstat/z_table.html.
the first display uses the z-score.
when using the z-score, the mean is 0 and the standard deviation is 1.
the raw score being compared to the mean is the z-score which is equal to 1.048427394.
in order to get that, however, you needed to calculate the standard error of the proportion which is equal to sqrt(p * q / n).
in that formula, p = .75, q = 1 - p = .25, n = 229 which resulted in s = .0286142848.
you then used the standard error to find the z-score using the formula of z = (x - m) / s where x was .78 and m was .75 and s was .0286142848.
the second display uses the raw score.
when using the raw score, the mean is .75 and the standard deviation is .0286142848.
the raw score being compared to the mean is .78, which is .03 higher than .75, or 3% higher than 75%.
in the first case, you needed to calculate the standard error and the z-score.
the standard error is the standard deviation of the proportion
in the second case, you need to calculate the standard error which is the standard deviation of the proportion.
here's the displays.
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