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The t-statistic is a fundamental concept in statistics that helps us compare the sample mean to the population mean. It's calculated using the formula: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] Here:

• \( \bar{x} \) is the sample mean, which in this case is 4.11.
• \( \mu_0 \) is the hypothesized population mean, which is 4.0.
• \( s \) represents the sample standard deviation of 1.43.
• \( n \) is the sample size, which was 25 in this instance.

By plugging these values into the formula, we get: \[ t = \frac{4.11 - 4.0}{1.43 / \sqrt{25}} = \frac{0.11}{0.286} \approx 0.385 \] This t-value, approximately 0.385, tells us how many standard deviations our sample mean is from the hypothesized population mean. If the value of the t-statistic is large, it suggests a more significant difference between the sample and population mean.

The P-value helps us determine the significance of our test result. It tells us the probability of obtaining a sample mean as extreme as the observed one, if the null hypothesis is true.To find the P-value, you first need to calculate the degrees of freedom, which is one less than the sample size: \( df = n - 1 = 24 \).Using statistical software or a t-distribution table, we can determine the P-value associated with our calculated t-statistic of 0.385 and 24 degrees of freedom. Given it is a two-tailed test, we find the probability of a value as extreme as our test statistic, in both tails of the distribution.In our example, you would find the P-value to be greater than 0.7, indicating weak evidence against the null hypothesis. A large P-value implies that there is a high probability of observing our sample mean if the null hypothesis is true, suggesting no strong evidence to reject the null hypothesis.

A confidence interval gives us a range of values within which the true population mean is likely to lie. For a 95% confidence interval, we express the range with a confidence level that suggests we are 95% confident the true mean falls within these bounds.The formula to calculate this interval is: \[ \bar{x} \pm t^* \frac{s}{\sqrt{n}} \]Where \( t^* \) is the critical value for a t-distribution at the desired confidence level and appropriate degrees of freedom. For our example, \( t^* \approx 2.064 \) when \( df = 24 \).Calculate the margin of error:\[ ME = 2.064 \times \frac{1.43}{\sqrt{25}} \approx 0.591 \] Thus, the confidence interval for the sample mean of 4.11 is:\[ (4.11 - 0.591, 4.11 + 0.591) = (3.519, 4.701) \] Rounding to one decimal place gives us (3.5, 4.7), which shows a range where the true population mean is likely to exist.

Sample size significantly impacts statistical measures such as the P-value and the confidence interval.

• **Effect on P-value:** A larger sample size typically suggests a smaller variability in results. This reduction in variability can result in a smaller P-value if the sample mean differs from the hypothesized mean, thus providing stronger evidence against the null hypothesis.
• **Effect on Confidence Interval:** With a larger sample size, the confidence interval becomes narrower. This indicates a more precise estimate of the population mean. A narrower interval means we can be more certain about the range in which the population mean lies.

In this example, compare the 25-sample size results with those from the larger sample of 1933. The larger sample size results in a smaller P-value and narrower confidence interval, demonstrating greater statistical power and precision.