Question

A successful waffle-man has recently developed a new recipe for waffles. To test the popularity of this new waffle compared to two other tried-and-true types of waffles, our friend the waffle-man randomly selected 180 lucky customers to vote on which of the three waffle types they liked best. Exactly 35% of these customers (or 63 in total) voted in favor of the new waffle. If all waffles were equally tasty, then the waffle-man knows to expect that each waffle would receive around 1/3 of the votes (so around 60 votes per waffle).

Are 63 votes for the new waffle enough to conclude that significantly more customers like it compared to the others? Luckily, our friend the waffle-man triple-majored in waffles, statistics, and clinical neurophysiology and knows how to objectively answer this question. He conducts a hypothesis test for proportions,

H0:p=0.333 H0:p=0.333, Ha:p>0.333 Ha:p>0.333

with a sample proportion of 63/180.

In carrying out this test, what null distribution should she use? In other words, what is the distribution of the test statistic assuming the null hypothesis is true? (Be sure to use at least four decimal places in your calculations.)

Select one:

a. A normal distribution centered at 1/3 with a standard deviation of about 0.0351.

b. A normal distribution centered at 0.333 with standard deviation of about 0.0356

c. A normal distribution centered at 0.35 with standard deviation of about 0.0356.

d. A normal distribution centered at 0.35 with standard deviation of about 0.0351.

e. We cannot use a null distribution in this problem because the population is not normally distributed.

Our friend the waffle-man is back and wants to do more hypothesis tests for proportions, but this time for four waffle recipes. He randomly selected 250 waffle consumers and found that 100 (40%) of the 250 preferred Waffle No. 2. He conducted a hypothesis test with

H0:p=0.25H0:p=0.25, Ha:p>0.25Ha:p>0.25.

Notice the proportion under the null distribution is p_0 = 0.25. The test statistic for this problem is 5.47. You can verify for yourself that the probability of observing this test statistic is nearly zero assuming the null hypothesis is true.

Now suppose we wish to conduct the same hypothesis test again if the true proportion is 0.35. In other words, we happen to know the true parameter value is 0.35, something that is typically not known. How does the test statistic change with this new information? What is the resulting p-value?

Hint: Try writing what the null hypothesis and test statistic would be given this new information. What would change, if anything?

Select one:

a. The test statistic becomes 1.61 with a p-value of 0.537.

b. The test statistic becomes 1.66 with a p-value of 0.095.

c. The test statistic becomes -3.23 with a p-value of 0.9995.

d. The test statistic does not change from its original value of 5.47, and the associated p-value does not change.

e. None of the other answers are correct.

Answer 1