A statisitics researcher collects data with a sample of 225 people indicating 170 of the perfer Nike athletic shoes over Asics athletic shoes.
a. what is the 90% confidence interval for the proportion of people who prefer Nike ocer Asics?
b. Which interpretation of that confidence intercal is best? And what is the reasoning?
1. "we think that 90% of allpeople prefer Nike over Asics"
2." we are 90% sure that 225 were in this sample"
3." we think this CI is one of the 90% of all confidence intervals that contain the true calue of p."
4. we confidently think that 170% of 225 people were sampled"

When you increase the confidence level but leave the number in the sample the same, the confidence interval increases.

When you increase the number on the sample but leave the confidence level alone, then the
confidence interval decreases.

Your 90% CI would be .709\leq{p}\leq{.803}

Because the margin of error is .047 and p=0.756

0.756-.047=.709

.756+.047=.803

If you increase the confidence level to 95%, you get a margin of error of 0.056.

See how it increased?

If you leave the confidence level at 90%, but increase the sample size to, say, 190 the CI

is .040. See how it decreased?

I have a wonderful Excel macro program I use for these problems. Going through these laborious calculations by hand with the cumbersome formulas is like calculating a square root by hand... rather obsolete. Most calculators perform lots of stats calculations. A TI-83 does just about everything if you learn how to use it.