Question 557:

Asked on December 16, 2008

Tags: Confidence Intervals , Margin of Error

1

Answer:

No answer provided yet.1. Lets first find the components of the population proportion so we can build the margin of error. We're told that 38% of 7100 said they watch network programs or .38*7100 = 2698 said they did. We'll use this number in the final step.
2. The margin of error for a proportion is found by multiplying the standard error of the proportion times a critical value from the normal distribution for the confidence level. The confidence level we're given is 95%. So we need the z-score which accounts for 95% of the area under the normal curve. We can find this in a z-table in the back of a statistics book or using the percentile to z-score calculator and entering .95 2-sided. We get 1.96.  So that's the first part.
3. Now we need to find the standard error of the proportion (SEP) which we can find as the standard deviation divided by the square root of the sample size. For a proportion, the standard deviation is found by taking the square root of p*q where p is .38 and q is 1-.38 or .62. So the standard deviation = SQRT(.38*.62) = .48539. The square root of the sample size is SQRT(7100) = 84.2615. So we divide these two values .48539/84.2615 = .00576, which is our SEP.  The margin of error is then .00576*1.96 = .011291.
4. If we wanted to build the confidence interval we'd just add and subtract the margin of error to the proportion .38+.011291 and .38-.011291 getting us a 95% CI between .3687 and .3913 of viewers watch network news.
5. We can double check our answers by using the confidence interval around a proportion calculator and entering 2698 for "passed" and 7100 for "total tested" and we get .0113 in the Margin of Error field, so we're right!