According to a recent marketing campaign, 90 drinkers of either Diet Coke or Die

Question

According to a recent marketing campaign, 90 drinkers of either Diet Coke or Diet Pepsi participated in a blind taste test to see which of the drinks was their favorite. In one Pepsi television commercial, an anouncer states that "in recent blind taste tests, more than one half of the surveyed preferred Diet Pepsi over Diet Coke." Suppose that out of those 90, 59 preferred Diet Pepsi. Test the hypothesis, using =0.01 that more than half of all participants will select Diet Pepsi in a blind taste test by giving the following:

Explanation / Answer

  Hypothesis Test for proportions:

Let X be the number of success in n independent and identically distributed Bernoulli trials, i.e., X ~ Binomial(n, p)

To test the null hypothesis of the form
H0: p = p0, or
H0: p p0, or
H0: p p0

Assuming that n*p0 > 10 and n * (1-p0) > 10 (some will say the necessary condition here is > 5, I prefer this more conservative assumption so that the approximations in the tail of the distribution are more accurate) then

find the test statistic z = (pHat - p0) / sqrt(p0 * (1-p0) / n)

where pHat = X / n

The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.
H1: p p0; p-value is the area in the tails greater than |z|
H1: p < p0; p-value is the area to the left of z
H1: p > p0; p-value is the area to the right of z

If the p-value is less than or equal to the significance level , i.e., p-value , then we reject the null hypothesis and conclude the alternate hypothesis is true. If the p-value is greater than the significance level, i.e., p-value > , then we fail to reject the null hypothesis and conclude that the null is plausible. Note that we can conclude the alternate is true, but we cannot conclude the null is true, only that it is plausible.

The hypothesis test in this question is:

H0: p 0.5 vs. H1: p > 0.5

The test statistic is:
z = ( 0.4555556 - 0.5 ) / ( ( 0.5 * (1 - 0.5 ) / 90 )
z = -0.843274

The p-value = P( Z > z )
= P( Z > -0.843274 )
= 0.8004624

Since the p-value is greater than the significance level of 0.05 we fail to reject the null hypothesis and conclude p 0.5 is plausible.

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