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When conducting a t-test, one of the key figures is the *sample size*, which refers to the number of observations or data points collected in a study. In the context of hypothesis testing, the sample size is crucial because:

• It influences the degrees of freedom.
• It affects the precision and reliability of the test results.
• Larger sample sizes usually lead to more accurate estimations of the population parameters.

For example, in the provided exercise, the sample size can be calculated using the degrees of freedom. Given that the degrees of freedom ( (df = 71), we determine the sample size ( ) by adding 1: = 71 + 1 = 72. Hence, in this case, the sample includes 72 batteries. Understanding how to determine the sample size is essential as it provides the foundation for calculating other statistical figures.

In a t-test, the term "degrees of freedom" refers to the number of independent values or observations that are free to vary when computing a statistic. It is an indicator of the sample size used in the statistical analysis.
Degrees of freedom can be calculated as:

• Degrees of freedom (df) = - 1
• For one-sample t-tests, the formula is particularly straightforward.

Given the example, where (df given is 71, we can trace back to find the actual sample size of 72 by applying the formula: = df + 1. This concept helps to determine how much information operates freely within the data set and is a critical part of hypothesis testing and ensuring the validity of a statistical test.

The test statistic in a t-test is a calculated value that allows us to test a hypothesis about the population means. It helps determine whether to reject the null hypothesis. This value is computed by comparing the sample mean to the hypothesized population mean, scaled by the standard deviation and square root of the sample size. In the example problem, the test statistic is given as (t = -1.09). This negative value suggests that the sample mean is slightly less than the hypothesized mean of 42.5 hours. Despite being lower, whether this difference is significant enough to reject the null hypothesis depends on the subsequent analysis involving the p-value. Understanding the test statistic's role helps quantify the difference between the expected and observed data.

The p-value is a critical concept in statistics, often used to determine the significance of the test results. It measures the probability that the observed data would occur if the null hypothesis were true.

• A low p-value (typically < 0.05) suggests we reject the null hypothesis, indicating a statistically significant difference.
• A high p-value implies insufficient evidence to reject the null hypothesis.

In the problem posed, the p-value is 0.139, which is higher than the usual significance level of 0.05. This result suggests that the observed difference between the sample mean and the claimed mean isn't statistically significant.
Thus, the null hypothesis, which states that the battery life lasts at least 42.5 hours, cannot be rejected based on the sample evidence. The p-value provides a clear numerical interpretation of our test results, assisting in making data-driven decisions with greater confidence.