In this chapter we introduce the idea of hypothesis testing in general, and then we look at the specifics for a hypothesis test for a single population proportion.

The idea of hypothesis testing is to use a single, properly collected sample to test whether or not an assumed value for some population parameter is actually true. We will make use of the sampling distributions that we have been using in the last couple of chapters.

Write the hypotheses under investigation for each of the following scenarios. (a)The Mars Candy Co. claims that since the introduction of blue as a color for M&M’s, 15% of all M&M’s produced are blue. In a randomly collected sample of 80 M&M’s, only 11 were blue. Is this strong enough evidence to claim that the true proportion of all M&M’s that are blue is less than 15%?

Write the hypotheses under investigation for each of the following scenarios. (b)In October 2000, the results of a large survey showed that 84.9% of males and 88.1% of females had graduated from high school. Does this study support the claim that females were more likely to graduate from high school than males in 2000?

Write the hypotheses under investigation for each of the following scenarios. (c)The posted speed limit on a certain residential road is 30mph. The residents believe that drivers are speeding on this road on average. They observe 20 randomly selected drivers on this road and find the mean speed to be 31.8mph with a standard deviation of 4.2mph. Is the residents’ belief accurate?

Write the hypotheses under investigation for each of the following scenarios. (d)Are the prices charged for a used camera higher on average when buying from a stranger than when buying from a friend?

Write the hypotheses under investigation for each of the following scenarios. (e)Mars Candy Co. claims that of all its peanut M&M’s, 20% are yellow, 20% are red, 10% each of orange, blue, and green, and the remaining were brown. Does the following data from a randomly selected sample support this claim?

A large P-value would say that, assuming H 0 is true, the sample is not that unusual. This would not support the alternative hypothesis, and so our conclusion would be “fail to reject H 0 ”. A small P-value would say that, assuming H 0 is true, the sample is unusual. Such a sample would lend support to the alternative hypothesis. Thus our conclusion would be “reject H 0 ”.

Hypotheses H 0 : p = # H a : one of (a) p > # (upper tail test) (b) p < # (lower tail test) (c) p ≠ # (two-tail test)

Test Statistic where p is the hypothesized value in H 0

P-value Depends on the alternative hypothesis: (a)(upper tail test) (b)(lower tail test) (c) (two-tailed test)

Validity/Assumptions We have a properly collected, random sample Sample size is not more than 10% of the population large sample size:

The Mars Candy Co. claims that since the introduction of blue as a color for M&M’s, 15% of all M&M’s produced are blue. In a randomly collected sample of 80 M&M’s, only 11 were blue. Is this strong enough evidence to claim that the true proportion of all M&M’s that are blue is less than 15%?

Press, go over to “Tests”, then scroll down and select “1-PropZTest…” Example 2 via the calculator

The drug Prevnar is a vaccine meant to prevent meningitis. In clinical trials, the vaccine was given to 710 randomly selected children 12 – 15 months in age. Of these, 96 experienced appetite loss as a side effect. Is this significant evidence that more than 10% of all children (age 12 – 15 months) that take Prevnar will have loss of appetite as a side affect? Test the relevant hypotheses at the  = 0.05 level.

We can replace the test statistic and P-value with a confidence interval for p calculated from the sample. If the hypothesized value is not in the interval, then we reject H 0 If the hypothesized value is in the interval, then we fail to reject H 0 All other “pieces” of the hypothesis test are the same.

The drug Prevnar is a vaccine meant to prevent meningitis. In clinical trials, the vaccine was given to 710 randomly selected children 12 – 15 months in age. Of these, 96 experienced appetite loss as a side effect. Is this significant evidence that more than 10% of all children (age 12 – 15 months) that take Prevnar will have loss of appetite as a side affect? Test the relevant hypotheses using a 95% confidence interval.