Hypothesis testing question -- how to define the null hypothesis to draw a conclusion

Question

Here is a hypothesis testing question I'm trying to work out but am struggling with how to draw a conclusion, maybe I'm not defining the null hypothesis correctly.

The question is as follows: Test the hypothesis that at least 68% of a population went to the movies once in the past year with a significance level of 5%. We surveyed 4000 people and found 3012 did go at least once in the past year.

Here I defined the null hypothesis as this: at least 68% (p >= .68) of the population went to the movies in the past year. (so the alternative hypothesis is that less than 68% of the population went)

Assuming the null hypothesis to be true, I calculated the standard deviation of the population to be 0.0074.

So, the Z-score = ((3012/4000)-0.68))/0.0074 = 9.86.

Here the critical z-value is 1.65 (for the 5% significance level).

Here is where I'm stuck -- what does this tell me? Is there something I'm doing wrong where identifying the null hypothesis?

Answer 1

Your hypotheses are fine.

You have a one tailed test, but you don't seem to consider whether you should be rejecting large or small proportions.

Your problem is your rejection rule! See your critical value. Does it correspond to a large proportion or a small one?

Forget the calculations and just think a little about the logic -- what does it seem should happen? If someone claims at least 68% saw a movie and your sample showed more than 68% (say 80%) would you doubt their claim? If your sample only showed 20%, would you be inclined to doubt the claim now?

So should you be rejecting for very high proportions or very low ones?

(The correct calculations simply tell you exactly where to draw the cut-off, but you should already see which side it should go on.)

It may also help to draw a diagram of a standard normal density and the normal approximation for a sample proportion when p=0.68 (i.e. at the boundary of the hypothesis space):

A z-score of 1.65 would correspond to a proportion higher than 0.68. A z-score a bit less than 1.65 would still correspond to a proportion higher than 0.68. A z-score of 0.5 would still correspond to a proportion higher than 0.68. They're all consistent with the claim!

Note that the z-score for an observed proportion of 0.68 (2720 people in the sample of 4000) would be $0$. Should you conclude that the claim ("...at least 0.68 of the population..." etc) is false if you saw exactly 0.68?

So what would be a clear indication that the claim in the null hypothesis is false? A proportion so small (so much lower than 0.68) that any proportion at least that small would be very unlikely to occur even if the true proportion was as low as 0.68 would make us suspicious of the null.

We can see that we should reject for observed proportions that are clearly below 0.68; these have negative z-values.