Question

A new toy has been introduced into retail outlets of a large chain of toy stores. A random sample of twelve outlets showed the following results for the number of toys sold in the first week: 113, 102, 87, 69, 111, 93, 84, 98, 108, 89, 98, 95. Find the sample mean, sample variance, and the sampling distribution of the sample mean (i.e., explain the nature of the distribution with the expected value and standard deviation of the sample mean). Test the hypothesis that the population mean is at least 100 at the 5% level of significance. What assumptions are needed for conducting these two tests?

Answer 1

To find the sample mean, sum the toy sales data and divide by the sample size. The sample variance is found by summing squared differences from the mean, divided by one less than the number of observations. To test the hypothesis that the population mean is at least 100, a t-test is used assuming a normally distributed population or a sufficiently large sample size for the Central Limit Theorem to apply.

Step-by-step explanation:

To calculate the sample mean, simply add up all the numbers in the sample and divide the sum by the number of observations in the sample. For the data provided (113, 102, 87, 69, 111, 93, 84, 98, 108, 89, 98, 95), the sample mean (μ) is found by totaling these values and then dividing by 12.

The sample variance (σ^2) is determined by squaring the difference between each observation and the sample mean, summing these squared differences, and then dividing by one less than the number of observations in the sample (n-1), to get an unbiased estimate.

When describing the sampling distribution of the sample mean, we use the Central Limit Theorem. It states that for a sufficiently large sample size from a population with a mean (μ) and standard deviation (σ), the distribution of the sample means will be approximately normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size (n).

To test the hypothesis that the population mean is at least 100 (μ ≥ 100), we use a t-test when the population standard deviation is unknown and the sample size is small (n<30). A one-sample t-test will compare the sample mean to the hypothesized mean. We would calculate the t-statistic and compare this to a critical value from a t-distribution table or use a p-value approach to determine if the hypothesis should be rejected at the 5% significance level. The necessary assumption is that the data are from a normally distributed population, or the sample size is large enough for the Central Limit Theorem to apply.