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This is from my Carnegie Integrated Math 3 text.

The water company says more people prefer tap water than bottled water in a blind taste test of 120 people in which 41 chose the bottled water. Use a 95% confidence interval to determine if the water company's claim is probably accurate.

So, I calculate $\hat p$ as 34%. So, ($\sqrt{\hat p(1−\hat p)/120)} = .043.$ and the 95% confidence interval would be 2(.043)=.0862. The book shows us to calculate 1 std deviation and then multiply by 2, not 1.96 to get the 95% CI.

So, now the book says the range from 34% +/− 8.6% would represent a 95% CI for the population proportion. That's 25.4−42.6%.

Here is where I get confused. The book then says, the water company is probably correct that more people like tap water since 50% is NOT in the 95% CI! HUH?

My thought was that the water company is probably correct because 66% (the percentage of people who preferred tap water) is statistically significant or outside the range calculated above.

Can you assist in throwing some light on the statistically significant question. Do I always compare 50% to the interval calculated?

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"Always" is a very strong word - of course not always. 50% is used when you have two outcomes and you want to test the hypothesis that they are equally probable. Clearly you should not compare the 66% to the interval, because 66% is the percentage of "tap preferrers" while the interval applies to the percentage of "bottle preferrers". Other than that, the quote from the book is full of inaccuracies. They have a one-sided hypothesis - why use a two-sided interval? The standard deviation should use p=0.5 and result in 0.05 (rather than 0.043). They should compare 0.66 to 0.5+1.65 x 0.05 = 0.58. Although the null hypothesis is rejected in both cases (one- or two-sided hypothesis), it does not mean that the water company is "probably right" - all it means is that we could not prove otherwise. Claiming that the company is right involves some Bayesian beliefs.

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  • $\begingroup$ So, as I now understand it, the text is using a point estimate of the 41 sampled people who prefer the bottled water and get phat to be$0.34$. Then they construct a 95% CI using the $0.34$, $0.66$ and sampling size of 120 people. This gives a range of values in which you'd expect to find the the true population parameter 95% of the time. What was confusing was it would seem to me the text should state an hypothesis of MOST dont like or MOST do like or MOST like about the same if they are going to compare the range to $50$%. Would you agree? $\endgroup$ – user163862 Sep 12 '17 at 20:52

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