New York City was the first U.S. city to require that chain restaurants post calorie counts for all menu items. Shortly after the new law was implemented, the research team of Bollinger, Leslie, & Sorensen (2010) decided to test the law’s effectiveness. For more than a year, they gathered data on every transaction at Starbucks coffee shops in several cities. They determined a population mean of 247 calories in products purchased by customers at stores without calorie postings. Based on the range of 0 to 1208 calories in customer transactions, we estimate a standard deviation of approximately 201 calories, which we’ll use as the population standard deviation for this example.

The researchers also recorded calories for a sample in New York City after calories were posted on Starbucks menus. They reported a mean of 232 calories per purchase, a decrease of 6%. For the purposes of this example, we’ll assume a sample size of 1000. Here’s how to apply hypothesis testing when comparing a sample of customers at Starbucks with calories posted on their menus to the general population of customers at Starbucks without calories posted on their menus.

We’ll use the six steps of hypothesis testing to analyze the calorie data. These six steps will tell us if customers visiting a Starbucks with calories listed on the menu consume fewer calories, on average, than customers visiting a Starbucks without calories listed on the menu. In fact, we use the six-step approach so often in this book that it won’t be long before it becomes an automatic way of thinking for you. Each step in the example below is followed by a summary that models how to report hypothesis tests.

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STEP 1: Identify the populations, distribution, and assumptions.

First, we identify the populations, comparison distribution, and assumptions, which help us to determine the appropriate hypothesis test. The populations are (1) all customers at those Starbucks with calories posted on the menu (whether or not the customers are in the sample) and (2) all customers at those Starbucks without calories posted on the menu. Because we are studying a sample rather than an individual, the comparison distribution is a distribution of means. We compare the mean of the sample of 1000 people visiting those Starbucks that have calories posted on the menu (selected from the population of all people visiting those Starbucks with calories posted) to a distribution of all possible means of samples of 1000 people (selected from the population of all people visiting those Starbucks that don’t have calories posted on the menu). The hypothesis test will be a z test because we have only one sample and we know the mean and the standard deviation of the population from the published norms.

Now let’s examine the assumptions for a z test. (1) The data are on a scale measure, calories. (2) We do not know whether sample participants were selected randomly from among all people visiting those Starbucks with calories posted on the menu. If they were not, the ability to generalize beyond this sample to other Starbucks customers would be limited. (3) The comparison distribution should be normal. The individual data points are likely to be positively skewed because the minimum score of 0 is much closer to the mean of 247 than it is to the maximum score of 1208. However, we have a sample size of 1000, which is greater than 30; so based on the central limit theorem, we know that the comparison distribution—the distribution of means—will be approximately normal.

Summary: Population 1: All customers at those Starbucks that have calories posted on the menu. Population 2: All customers at those Starbucks that don’t have calories posted on the menu.

The comparison distribution will be a distribution of means. The hypothesis test will be a z test because we have only one sample and we know the population mean and standard deviation. This study meets two of the three assumptions and may meet the third. The dependent variable is scale. In addition, there are more than 30 participants in the sample, indicating that the comparison distribution will be normal. We do not know whether the sample was randomly selected, however, so we must be cautious when generalizing.

STEP 2: State the null and research hypotheses.

Next we state the null and research hypotheses in words and in symbols. Remember, hypotheses are always about populations, not samples. In most forms of hypothesis testing, there are two possible sets of hypotheses: directional (predicting either an increase or a decrease, but not both) or nondirectional (predicting a difference in either direction).

The first possible set of hypotheses is directional. The null hypothesis is that customers at those Starbucks that have calories posted on the menu do not consume fewer mean calories than customers at those Starbucks that don’t have calories posted on the menu; in other words, they could consume the same or more mean calories, but not fewer. The research hypothesis is that customers at those Starbucks that have calories posted on the menu consume fewer mean calories than do customers at Starbucks that don’t have calories posted on the menu. (Note that the direction of the hypotheses could be reversed.)

The symbol for the null hypothesis is H₀. The symbol for the research hypothesis is H₁. Throughout this text, we use μ for the mean because hypotheses are about populations and their parameters, not about samples and their statistics. So, in symbolic notation, the hypotheses are:

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H₀: μ₁ ≥ μ₂

H₁: μ₁ < μ₂

For the null hypothesis, the symbolic notation says that the mean calories consumed by those in population 1, customers at those Starbucks with calories posted on the menu, is not lower than the mean calories consumed by those in population 2, customers at those Starbucks without calories posted on the menu. For the research hypothesis, the symbolic notation says that the mean calories consumed by those in population 1 is lower than the mean calories consumed by those in population 2.

This hypothesis test is considered a one-tailed test. A one-tailed test is a hypothesis test in which the research hypothesis is directional, positing either a mean decrease or a mean increase in the dependent variable, but not both, as a result of the independent variable. One-tailed tests are rarely seen in the research literature; they are used only when the researcher is absolutely certain that the effect cannot go in the other direction or the researcher would not be interested in the result if it did.

The second set of hypotheses is nondirectional. The null hypothesis states that customers at Starbucks with posted calories (whether in the sample or not) consume the same number of calories, on average, as customers at Starbucks without posted calories. The research hypothesis is that customers at Starbucks with posted calories (whether in the sample or not) consume a different average number of calories than do customers at Starbucks without posted calories. The means of the two populations are posited to be different, but neither mean is predicted to be lower or higher.

The hypotheses in symbols would be:

H₀: μ₁ = μ₂

H₁: μ₁ ≠ μ₂

For the null hypothesis, the symbolic notation says that the mean number of calories consumed by those in population 1 is the same as the mean number of calories consumed by those in population 2. For the research hypothesis, the symbolic notation says that the mean number of calories consumed by those in population 1 is different from the mean number of calories consumed by those in population 2.

This hypothesis test is considered a two-tailed test. A two-tailed test is a hypothesis test in which the research hypothesis does not indicate a direction of the mean difference or change in the dependent variable, but merely indicates that there will be a mean difference. Two-tailed tests are much more common than are one-tailed tests. We will use two-tailed tests throughout this book unless we tell you otherwise. If a researcher expects a difference in a certain direction, he or she might have a one-tailed hypothesis; however, if the results are in the opposite direction, the researcher cannot then switch the direction of the hypothesis.

Summary: Null hypothesis: Customers at those Starbucks that have calories posted on the menu consume the same number of calories, on average, as do customers at Starbucks that don’t have calories posted on the menu—H₀: μ₁ = μ₂. Research hypothesis: Customers at those Starbucks that have calories posted on the menu consume a different number of calories, on average, than do customers at those Starbucks that don’t have calories posted on the menu—H₁: μ₁ ≠ μ_2.

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STEP 3: Determine the characteristics of the comparison distribution.

Now we determine the characteristics that describe the distribution with which we will compare the sample. For z tests, we must know the mean and the standard error of the population of scores; the standard error for samples of this size is calculated from the standard deviation of the population of scores. Here, the population mean for the number of calories consumed by the general population of Starbucks customers is 247, and the standard deviation is 201. The sample size is 1000. Because we use a sample mean in hypothesis testing, rather than a single score, we must use the standard error of the mean instead of the population standard deviation (of the scores). The characteristics of the comparison distribution are determined as follows:

μ_M = μ = 247

Summary: μ_M = 247; σ_M = 6.356

STEP 4: Determine the critical values, or cutoffs.

Next we determine the critical values, or cutoffs, to which we can compare the test statistic. As stated previously, the research convention is to set the cutoffs to a p level of 0.05. For a two-tailed test, this indicates the most extreme 5%—that is, the 2.5% at the bottom of the comparison distribution and the 2.5% at the top. Because we calculate a test statistic for the sample—specifically a z statistic—we report cutoffs in terms of z statistics. We use the z table to determine the scores for the top and bottom 2.5%.

We know that 50% of the curve falls above the mean, and we know that 2.5% falls above the relevant z statistic. By subtracting (50% – 2.5% = 47.5%), we determine that 47.5% of the curve falls between the mean and the relevant z statistic. When we look up this percentage on the z table, we find a z statistic of 1.96. So the critical values are –1.96 and 1.96 (Figure 7-10).

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Summary: The cutoff z statistics are –1.96 and 1.96.

STEP 5: Calculate the test statistic.

In step 5, we calculate the test statistic, in this case a z statistic, to find out what the data really say. We use the mean and standard error calculated in step 3:

Summary:

STEP 6: Make a decision.

Finally, we compare the test statistic to the critical values. We add the test statistic to the drawing of the curve that includes the critical z statistics (Figure 7-11). If the test statistic is in the critical region, we can reject the null hypothesis. In this example, the test statistic, –2.36, is in the critical region, so we reject the null hypothesis. An examination of the means tells us that the mean number of calories consumed by customers at those Starbucks with calories posted on the menu is lower than the mean number of calories consumed by customers at those Starbucks with no calories posted. So, even though we had nondirectional hypotheses, we can report the direction of the finding—that is, it appears that customers consume fewer calories, on average, at Starbucks that post calories on the menu than at Starbucks that do not post calories on the menu.

If the test statistic is not beyond the cutoffs, we fail to reject the null hypothesis. This means that we can only conclude that there is no evidence from this study to support the research hypothesis. There might be a real mean difference that is not extreme enough to be picked up by the hypothesis test. We just can’t know.

Summary: We reject the null hypothesis. It appears that fewer calories are consumed, on average, by customers at Starbucks that post calories on the menus than by customers at Starbucks that do not post calories on the menu.

The researchers who conducted this study concluded that the posting of calories by restaurants does indeed seem to be beneficial. The 6% reduction may seem small, they admit, but they report that the reduction was larger—a 26% decrease in calories—among those consuming 250 or more calories per visit and among those making food purchases. Also, the researchers theorized that given data such as these, chains might respond by adding lower-calorie choices, leading to further reductions in average calories consumed.

Solutions to these Check Your Learning questions can be found in Appendix D.