A company selling licenses for new e-commerce computer...

A company selling licenses for new e-commerce computer software advertises that firms using this software obtain, on average during the first year, a yield of $10 \%$ on their initial investments. A random sample of 10 of these franchises produced the following yields for the first year of operation: $\begin{array}{llllllllll}6.1 & 9.2 & 11.5 & 8.6 & 12.1 & 3.9 & 8.4 & 10.1 & 9.4 & 8.9\end{array}$ Assuming that population yields are normally distributed, test the company's claim.

A company selling licenses for new e-commerce computer software advertises that firms using this software obtain, on average during the first year, a yield of $10 \%$ on their initial investments. A random sample of 10 of these franchises produced the following yields for the first year of operation:
$\begin{array}{llllllllll}6.1 & 9.2 & 11.5 & 8.6 & 12.1 & 3.9 & 8.4 & 10.1 & 9.4 & 8.9\end{array}$
Assuming that population yields are normally distributed, test the company's claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a stated null hypothesis in favor of an alternative hypothesis. This process involves comparing a test statistic, calculated from sample data, to a critical value from a probability distribution, thereby providing a formal mechanism for decision-making under uncertainty.

Null and Alternative Hypotheses

In any hypothesis test, the null hypothesis (H?) typically represents a claim of no difference or no effect, while the alternative hypothesis (H?) represents the claim that contradicts the null. The formulation of these hypotheses is crucial as it establishes the framework for statistical testing, determining what constitutes evidence against the claim stated in the null hypothesis.

One-Sample t-Test

The one-sample t-test is a statistical test used to compare the sample mean to a known or claimed population mean, especially when the population standard deviation is unknown and the sample size is small. It relies on the t-distribution, which accounts for the additional uncertainty introduced by estimating the standard deviation from a limited sample.

Test Statistic

The test statistic in the context of a one-sample t-test is calculated by subtracting the hypothesized population mean from the sample mean and then dividing the result by the standard error of the mean. This standard error is the sample standard deviation divided by the square root of the sample size. The computed test statistic quantifies the deviation of the sample from what is expected under the null hypothesis.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation that are free to vary. In the context of a one-sample t-test, the degrees of freedom are typically the sample size minus one. This parameter is important because it influences the shape of the t-distribution, which is used to determine critical values for the test.

Assumptions of Normality

The assumption of normality states that the data should be approximately normally distributed, especially when the sample size is small. This assumption is particularly important for many parametric tests, including the one-sample t-test, because the validity of the test results depends on the data conforming to the underlying assumptions of the statistical model.

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