Egolf.com receives an average of 6.5 returns per day from...

Egolf.com receives an average of 6.5 returns per day from online shoppers. For a sample of 12 days, they received the following number of returns. $$\begin{array}{llllllllllll}0 & 4 & 3 & 4 & 9 & 4 & 5 & 9 & 1 & 6 & 7 & 10\end{array}$$ At the .01 significance level, can we conclude the mean number of returns is less than $6.5 ?$

Egolf.com receives an average of 6.5 returns per day from online shoppers. For a sample of 12 days, they received the following number of returns.
$$\begin{array}{llllllllllll}0 & 4 & 3 & 4 & 9 & 4 & 5 & 9 & 1 & 6 & 7 & 10\end{array}$$
At the .01 significance level, can we conclude the mean number of returns is less than $6.5 ?$

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent values that are free to vary in the calculation of a statistic. In a one-sample t-test, the degrees of freedom are calculated as the sample size minus one. This value is used to determine the critical value from the t-distribution, which is essential for hypothesis testing.

One-Sample t-Test

The one-sample t-test is used when the sample size is small and the population standard deviation is unknown. It compares the sample mean against a known value to determine if there is a significant difference, under the assumption that the sample is drawn from a normally distributed population.

Hypothesis Testing

Hypothesis testing is the framework used to decide whether there is enough statistical evidence in a sample to infer that a certain condition holds for the entire population. In this context, it involves setting up a null hypothesis (commonly representing the status quo or a specific claim) and an alternative hypothesis (representing a contradicting claim), then using sample data to determine which hypothesis is more likely to be true.

Null and Alternative Hypotheses

The null hypothesis usually asserts that there is no significant difference or that the parameter of interest equals a specified value (in this case, a mean of 6.5). The alternative hypothesis posits that the true mean is less than the specified value. Structuring these hypotheses correctly is critical because it influences the choice of test, interpretation of results, and the direction of the statistical test.

Significance Level

The significance level (alpha) defines the threshold for rejecting the null hypothesis. It is the probability of making a Type I error (rejecting a true null hypothesis). In this scenario, an alpha of 0.01 indicates a strong requirement for evidence before concluding that the mean number of returns is less than 6.5.

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