In Exercises 3–6, determine whether a normal sampling distribu...

Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p > 0.70, α=0.04. Sample statistics: p_hat = 0.64, n=225

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a researcher claims that the proportion of adults who exercise regularly is greater than 0.55. In a random sample of N equal to 180 adults, we have that PHAT is equal to 0.50. Can a normal sampling distribution be used? If so, test the claim at alpha equal to 0.05. Now for the first part of the problem, we need to determine if a normal sampling distribution could be used. And for this, we need to calculate two quantities. The first quantity that we need to calculate is going to be n multiplied by P, where N here is going to be equal to 180 and P is going to be equal to 0.55. So substituting in these values, this is going to be equal to 180, multiplied by 0.55, and evaluating the expression, this is going to be equal to 99. The other quantity that we need to calculate is N, multiplied by 1 minus P in quantity. And again, substituting into the values, this is going to be equal to 180, multiplied by the quantity of 1 minus 0.55. In quantity, and when we evaluate this expression, this is going to be equal to 81. Now, both of these quantities are actually greater than 5, so that means that a normal sampling distribution can be used. So the answer to the first part of this problem is going to be yes. A normal sampling distribution can be used. Now, for the second part, we need to test the claim at alpha equal to 0.05. So we're going to begin by writing down our hypotheses. Now, for our null hypothesis, H0, we're going to have that P is less than or equal to 0.55. And our alternative hypothesis, HA is going to be that P is greater than 0.55, which is the claim that we were given in the problem. Now, since we have a graded ensign in our alternative hypothesis, that means that our hypothesis test is going to be right tailed. So, next we need to calculate our test statistics, Z, and here we call your formula 4 Z. That's gonna be Z, it's going to be equal to the quantity of Phat minus P. In quantity divided by the square root of the quantity of p multiplied by the quantity of 1 minus p in quantity divided by n in quantity, and we have to have values for all those quantities, and so we'll substitute in those values. So Z is going to be equal to the quantity for Phat, we have 0.50 minus our value for P, which is going to be 0.55. In quantity, and this can be divided by the square root of the quantity of 0.55 multiplied by the quantity of 1 minus 0.55 in quantity divided by 180. In quantity. And so when we evaluate this expression, this is going to be approximately equal to minus 1.348. So next we need to find our critical Z value. So next we need to look up our critical value that corresponds to alpha equal to 0.05. So looking this up in our tables, we see that our critical value, Z 0.05, is going to be equal to. 1.645. So now we need to uh to compare our calculated Z value to our critical Z value. So, when we make the comparison, we see that our calculated Z value of Z is less than our critical Z value of Z 0.05. And since we're looking at a right tail test, that means that the rejection region is actually going to be greater than our critical value. So, in this case, since our calculated value is less than our critical value, that means that we fail to reject our null hypothesis. So we fail to reject. Each night And since we fail to reject our null hypothesis, that means that there's insufficient evidence to support our researcher's claim. So that means that there is insufficient evidence to support the claim that the proportion of adults who exercise regularly is greater than 0.55. So that's gonna be the answer to the second part of this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p > 0.70, α=0.04. Sample statistics: p_hat = 0.64, n=225

Key Concepts

Normal Sampling Distribution

Hypothesis Testing

Significance Level (α)

Watch next