I am confused solving this problem

Question

A real estate agent claims that 62% of all private residences being built today are 3-bedroom homes. To test this claim, a large sample of new residences is inspected; the proportion of these homes with 3 bedrooms is recorded and used as the test statistic. State the null and alternative hypotheses to be used in this test and determine the location of the critical region.

Answer 1

Hi Charles!

Here, it sounds like we want to test whether our test statistic, i.e., the proportion p of new residences with 3 bedrooms, is equal to 62% or not. In this case 62% is the "status quo," or our baseline belief since it's what the real estate agent claims to be true. This means that our null hypothesis (which is the status quo or baseline belief by definition) is:

H₀: p = 0.62

On the other hand, our alternative hypothesis is just that the null hypothesis isn't the case, or that the proportion of new residences with 3 bedrooms is NOT equal to 0.62. In particular, we notice that we aren't interested in the problem whether the proportion p is greater than 0.62 or less than 0.62, so we default to an alternative that simply looks at p not equal to 0.62. In other words, our alternative hypothesis is:

H_A: p ≠ 0.62. (Note that a different notation for the alternative hypothesis is H₁).

We can use these hypotheses to conduct a hypothesis test, in which we calculate a z-score, which is a test statistic:

z = (p – 0.62) / (square root of [0.62 x (1 – 0.62)/sample size]).

We can use this z-score we calculate as per above (which you don't actually need to do for this problem, by the way, but you'd need to do in general hypothesis testing question) and the significance level (which also isn't given to you in this problem) to find the critical region. The critical region lies in both tails, and how far into the tails it goes depends on the significance level. The lower the significance level, the closer to the center of the normal distribution the cutoffs are, and the higher the significance level, the further into the tails the cutoffs get pushed.

I don't think I can add a photo on here, but I'd recommend drawing out a normal distribution for yourself, and basically in this case, since it's a two sided test, the critical region would be both tails of that normal distribution. If our alternative hypothesis were just p > 0.62, then the critical region would only be the upper tail (the right tail), and if the alternative hypothesis were p < 0.62, then the critical region would only be the lower (left) tail.

To summarize, you can't actually calculate the critical region for this problem since you don't have enough information (in particular a significance level), but your critical region is just both tails of the normal distribution. This means you'd reject the null hypothesis for values that fall in either the upper tail or the lower tail (just by definition of a critical region).

Let me know if that makes sense and if you have more questions!

Best,

Andrea