Question

Problem: The manater of a large gym believes that the amount of exercise that the 1300 customers of the gym get, on average, per week is different from five hours. A random sample of 50 customers from the gym were surveyed with the following question: How many hours of exercise do you usually get per week? From the 50 respondents, the manager obtained a one-sample t-interval to estimate μ, the mean amount of exercise, in hours, that all customers of the gym get per week. Assume the conditions for inference have been met. At the 94 percent confidence level, the interval was 4.914 ± 0.503.

c) The manager wants to use an inference procedure to investigate whether the sample data provide convincing statistical evidence of a difference in the population mean amounts of exercise, in hours, that customers who prefer aerobic exercise and customers who prefer anaerobic exercise would get per week. Explain whether each condition for inference is met

Answer 1

In order to use an inference procedure to compare the mean weekly exercise time between customers preferring aerobic versus anaerobic exercise, conditions of randomness, normality, and independence must be satisfied. A random sample is required for both groups, and assuming the sample sizes are large enough, normality can be assumed under the Central Limit Theorem.

Step-by-step explanation:

The manager of a large gym is looking to investigate whether there is a significant difference in the mean amount of exercise between two groups within the gym's clientele: those who prefer aerobic exercise and those who prefer anaerobic exercise. To properly conduct such an inference procedure, certain conditions must be met:

• Randomness: The sample must be random, meaning that each customer of the gym has an equal chance of being selected for the study.
• Normality: The sampling distribution should be approximately normal or the sample size should be large enough that the Central Limit Theorem applies. For comparing means, this can often be assumed if each sample size is at least 30.
• Independence: The sampled customers must be independent of each other, which is usually safe to assume if the sample is less than 10% of the population and there is no reason to believe that the preferences of one customer would influence another.

Since the manager has a random sample of 50 customers out of 1300, it is likely that the randomness and independence conditions are satisfied. However, to proceed with the inference about the difference in means, the manager would need to gather two separate random samples: one from customers who prefer aerobic exercise, and another from customers who prefer anaerobic exercise. Each group should meet the sample size requirements to assume normality of the distribution of sample means.