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The t-statistic is a value derived from a t-test, which helps determine if there is a significant difference between a sample mean and a known population mean. It considers the sample size and variability to provide a measure of how far the sample mean is from the population mean, in terms of standard errors. To calculate the t-statistic, follow these steps:

• Compute the sample mean (\( \bar{x} \)).
• Find the sample standard deviation (\( s \)).
• Use the formula: \[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]where \( \mu \) represents the population mean and \( n \) is the sample size.

For instance, in our example, if the sample mean is 5.6375 and the sample standard deviation is 0.5123 with a sample size of 8, and the population mean (\( \mu \)) is 6, then the t-statistic is calculated as: \( t = \frac{5.6375 - 6}{0.5123/\sqrt{8}} = -2.518 \) This negative value indicates that the sample mean is less than the hypothesized population mean.

The p-value is a crucial part of hypothesis testing, offering a way to measure the strength of evidence against the null hypothesis. This probability score assesses whether the observed data contradicts the null hypothesis. When you calculate the p-value:

• You compare it to your significance level (commonly 0.01 or 0.05).
• If the p-value is less than the significance level, you reject the null hypothesis, suggesting that the alternative hypothesis may be true.
• If it's higher, you fail to reject the null hypothesis, implying there isn't strong enough evidence against it.

In our scenario, with a t-statistic of -2.518, the p-value was approximately 0.019. Since 0.019 is greater than the 0.01 significance level, this provides insufficient evidence to claim that the average mortgage rate is less than 6 percent.

The sample mean is the average value of a set of observations or data points from a sample. It serves as an estimate of the population mean, offering insight into the data's central tendency.To calculate the sample mean:

• Add all the sample data values together.
• Divide by the number of observations in the sample.

For example, given the mortgage rates from eight banks in our scenario: \( 4.8, 5.3, 6.5, 4.8, 6.1, 5.8, 6.2, 5.6 \)The sample mean (\( \bar{x} \)) is calculated as:\[ \bar{x} = \frac{4.8 + 5.3 + 6.5 + 4.8 + 6.1 + 5.8 + 6.2 + 5.6}{8} = 5.6375 \] This mean value is then used in further statistical analysis, such as in t-tests.

The sample standard deviation measures the dispersion or spread of a set of data points around the sample mean. It indicates how much the values in the sample deviate from the mean, which is important for understanding variability.To calculate the sample standard deviation:

• Find the deviation of each data point from the sample mean.
• Square each deviation.
• Sum these squared deviations.
• Divide the total by \( n - 1 \), where \( n \) is the sample size.
• Take the square root of the result.

In our example, with mortgage rate data:- Compute each squared deviation from the mean.- The sum of squared deviations was 3.0525.- With a sample size of 8, divide by 7 (because our degrees of freedom \( n-1 = 7 \)).The sample standard deviation is then:\[ s = \sqrt{\frac{3.0525}{7}} = 0.5123 \]A lower standard deviation indicates that the data points tend to be closer to the mean, while a higher standard deviation shows more spread.