Hypothesis Tests – Some Examples
Example 1: Directional hypothesis test for a population mean.
The U. S. Food and Drug Administration recommends that individuals consume 1000 mg of calcium daily. The International Dairy Foods Association (IDFA) sponsors and advertising campaign aimed at male teenagers to try to increase calcium consumption. After the campaign, the IDFA obtained a random sample of 50 male teenagers and found that the mean amount of calcium consumed was 1081 mg, with a standard deviation of s = 426 mg. Conduct a test to determine whether the campaign was effective. Use the = 0.05 level of significance.
We want to prove that the mean, , calcium consumption for male teenagers is greater than the FDA recommendation of 1000 mg per day. Hence,
Step 1: H0: 1000 mg.versusHa: > 1000 mg.
Note that the alternative hypothesis states what we are trying to prove; the null hypothesis states the opposite.
Step 2: n = 50, = 0.05
Note that in the real world, we would decide on a value for based on our examination of the possible consequences of the two possible types of mistakes we could make – a Type I error or a Type II error.
In this situation, a Type I error would be that we concluded that the campaign did convince male teenagers to increase their calcium consumption, when in fact it did not. Possible consequences might be that the IDFA would continue to fund the ad campaign, not realizing that it did not work.
A Type II error would be to fail to conclude that male teenagers were consuming enough calcium, when in fact they are. The IDFA might then stop its ad campaign, even though it would be effective in convincing male teenagers to consume enough calcium.
Step 3: Since we are testing a hypothesis about a population mean, the test statistic is , which under H0 has a t distribution with d.f. = 49.
Step 4: The alternative hypothesis says “greater than.” Therefore the rejection region is a right-hand tail of the t-distribution. The area of this right-hand tail is 0.05, our chosen significance level. The rejection region looks like:
Step 5: Now, we select the random sample from the population, collect the data, and do the calculations. We find that 1081 mg, s = 426 mg, t = 1.3445, and p-value = 0.0925. The p-value is greater than our chosen significance level.
Step 6: We fail to reject H0 at the 0.05 level of significance. We do not have sufficient evidence to conclude that the mean daily calcium consumption by male teenagers is greater than the FDA recommended amount of 1000 mg.
Example 2: Hypothesis test about a population proportion.
In 1995, 40% of adults aged 18 years or older reported that they had “a great deal” of confidence in the public schools. On June 1, 2005, the Gallup Organization released results of a poll in which 372 of 1004 adults aged 18 years or older stated that they had “a great deal” of confidence in the public schools. Does the evidence suggest that the proportion of adults aged 18 years or older having “a great deal” of confidence in the public schools has decreased between 1995 and 2005? Use = 0.05.
Since we are testing hypotheses about a population proportion, we are collecting the data using a binomial experiment. There are 1004 trials in the experiment. The trials are independent of each other due to random sampling. The trials are identical to each other because each trial consists of randomly selecting an adult and asking whether he/she has “a great deal” of confidence in the public schools. There are two possible outcomes for each trial: either the person says, “Yes” or the person says, “No.” Due to random sampling, the probability of success is the same for each trial, namely the fraction of the population who would say, “Yes.”
Step 1: H0: p 0.40versusHa: p < 0.40
Step 2: n = 1004, = 0.05
Step 3: The test statistic is , which under H0 has an approximate standard normal distribution.
Step 4: The rejection region is:
Step 5: Now, we select the random sample from the population, collect the data, and do the calculations. We find that , z = -1.9069, and p-value = 0.0283. The p-value is less than our chosen significance level.
Step 6: We reject H0 at the 0.05 level of significance. We have sufficient evidence to conclude that the fraction of adults 18 years or older who have “a great deal” of confidence in the public schools decreased between 1995 and 2005.