In Problems 3–6, use the results in the table to (b) determine the linear correlation between the observed values and expected z-scores, (c) determine the critical value in Table VI to assess the normality of the data.

Verified step by step guidance

1. Step 1: Understand the data provided. You have observed values, their corresponding relative frequencies \(f_i\), and expected z-scores. The goal is to analyze the linear correlation between the observed values and expected z-scores, and then assess normality using a critical value from Table VI.
2. Step 2: To determine the linear correlation coefficient \(r\) between the observed values and expected z-scores, use the formula for Pearson's correlation coefficient: \[r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\] where \(x_i\) are the observed values, \(y_i\) are the expected z-scores, \(\bar{x}\) is the mean of observed values, and \(\bar{y}\) is the mean of expected z-scores.
3. Step 3: Calculate the means \(\bar{x}\) and \(\bar{y}\) of the observed values and expected z-scores respectively. Then compute the deviations \((x_i - \bar{x})\) and \((y_i - \bar{y})\) for each pair, multiply these deviations, and sum them up. Also calculate the sums of squared deviations for both variables.
4. Step 4: Substitute these sums into the formula for \(r\) to find the correlation coefficient. This value will indicate how closely the observed values and expected z-scores are linearly related.
5. Step 5: To determine the critical value from Table VI for assessing normality, identify the sample size \(n\) (which is the number of data points), then look up the critical value corresponding to \(n\) and the chosen significance level (commonly 0.05). This critical value will be used to compare against the correlation coefficient to decide if the data significantly deviates from normality.

Key Concepts

Linear Correlation

Linear correlation measures the strength and direction of a linear relationship between two variables. It is quantified by the correlation coefficient, which ranges from -1 to 1. A value close to 1 indicates a strong positive linear relationship, while a value near 0 suggests no linear relationship. In this context, it helps assess how well observed values align with expected z-scores.

Expected z-scores

Expected z-scores represent standardized values assuming the data follows a normal distribution. They indicate how many standard deviations an observation is from the mean. Comparing observed values to expected z-scores helps evaluate if the data fits a normal distribution, which is essential for many statistical tests.

Critical Value and Normality Test

A critical value is a threshold from a statistical table used to decide whether to reject a null hypothesis. In normality tests, it helps determine if the data significantly deviates from a normal distribution. Comparing a test statistic to the critical value from Table VI allows assessment of the data's normality.