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Researchers investigated how the size of a bowl affects how much ice cream people tend to scoop when serving themselves. \({ }^{12}\) At an "ice cream social," people were randomly given either a 17 -oz or a 34 -oz bowl (both large enough that they would not be filled to capacity). They were then invited to scoop as much ice cream as they liked. Did the bowl size change the selected portion size? Here are the summaries: 12Brian Wansink, Koert van Ittersum, and James E. Painter, "Ice Cream Illusions: Bowls, Spoons, and Self-Served Portion Sizes," Am. J. Prev. Med. 2006 . Test an appropriate hypothesis and state your conclusions. For assumptions and conditions that you cannot test, you may assume that they are sufficiently satisfied to proceed.

Step by step solution

01

State the hypotheses

The null hypothesis (H0) is that bowl size does not affect the amount of ice cream servings: the mean amount scooped is the same for 17-oz and 34-oz bowls. The alternative hypothesis (Ha) is that bowl size does affect the amount of ice cream servings: the mean amount scooped is different between 17-oz and 34-oz bowls.

02

Check the conditions

Before carrying out the hypothesis test, check that the conditions for the test are met. In this case, the problem states that we assume meeting the conditions, such as the data being normally distributed or the samples being independent.

03

Perform the test

To perform the hypothesis test, compute the test statistic and the p-value using the data gathered about the amount of ice cream scooped into each bowl size. This depends on the exact data available, and may involve a t-test or a z-test procedure.

04

Make a decision and interpret the result

If the p-value is small (typically, less than 0.05), reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis, Ha. This means that the bowl size does have an effect on the ice cream serving size. If the p-value is large, do not reject H0. This means there is not enough evidence to conclude that bowl size has an effect on the ice cream servings.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In hypothesis testing, the null hypothesis, often denoted as \( H_0 \), is a statement that indicates no effect or no difference. It's a baseline proposition that assumes the absence of a relationship between variables. For instance, in our ice cream bowl size study, the null hypothesis posits that the capacity of the bowl (17-oz or 34-oz) has no impact on the amount of ice cream people serve themselves.

Alternative Hypothesis

Contrasting the null hypothesis, the alternative hypothesis \( H_a \) or \( H_1 \) asserts that there is a difference, effect, or relationship. In the context of the ice cream study, the alternative hypothesis claims that the size of the bowl does indeed influence the serving size. This hypothesis is what researchers aim to support, looking for evidence through experimental data that challenges the status quo established by the null hypothesis.

P-value

The p-value is a crucial concept in hypothesis testing. It represents the probability of observing results as extreme as those obtained, assuming the null hypothesis is true. A small p-value suggests that the observed data is unlikely under \( H_0 \), indicating that the alternative hypothesis may be true. If the p-value is low (commonly below 0.05), we might reject \( H_0 \) in favor of \( H_a \) with a certain level of confidence.

Statistical Significance

Statistical significance is an indication that the result of a test is unlikely to have occurred by chance alone. It's often determined by the p-value: if the p-value falls below a pre-determined threshold (like 0.05 or 0.01), the results are considered statistically significant. Significance, however, does not imply practical importance; it merely suggests there's evidence that the null hypothesis may not hold.

T-test

The t-test is a statistical test used to compare the means of two groups. It's especially useful when dealing with small sample sizes or when the population standard deviation is unknown. In our ice cream experiment, a t-test could be the appropriate test to compare the mean servings between the two bowl sizes, under the assumption that the data follows a normal distribution and variability in the two groups is similar.

Z-test

A z-test is another type of statistical hypothesis test, which is appropriate when the population variance is known and the sample size is large. It's similar to a t-test but uses a standard normal distribution to determine the test statistic. While the t-test is better suited for our ice cream serving size example due to likely unknown population variance, in instances where we have a large sample and known variance, a z-test would be the tool of choice.

Experimental Study Design

Experimental study design refers to the way a scientific study is structured to test a hypothesis. This design includes the random assignment of subjects to different groups, manipulation of independent variables, and control of confounding variables. The ice cream serving study likely used a simple randomized design to assign participants to receive either a 17-oz or a 34-oz bowl,balancing out potential confounders and aiming to isolate the effect of bowl size on serving portions.