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: Organize the data by listing the observed values and their corresponding expected z-scores from the table. This will help in calculating the correlation between these two variables.
2. Step 2: Calculate the mean of the observed values and the mean of the expected z-scores. Use the formulas: \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\) for observed values and \(\bar{z} = \frac{1}{n} \sum_{i=1}^n z_i\) for expected z-scores, where \(n\) is the number of data points.
3. Step 3: Compute the covariance between the observed values and expected z-scores using the formula: \(\text{Cov}(X,Z) = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(z_i - \bar{z})\).
4. Step 4: Calculate the standard deviations of the observed values and expected z-scores separately using: \(s_x = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}\) and \(s_z = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (z_i - \bar{z})^2}\).
5. Step 5: Determine the linear correlation coefficient \(r\) by dividing the covariance by the product of the standard deviations: \(r = \frac{\text{Cov}(X,Z)}{s_x s_z}\). This value measures the strength and direction of the linear relationship between observed values and expected z-scores.
6. Step 6: To assess normality, refer to Table VI (critical values table for correlation coefficients) using the sample size \(n\) and a chosen significance level (commonly 0.05). Compare the calculated correlation coefficient \(r\) to the critical value from the table 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, typically represented by the correlation coefficient (r). Values range from -1 to 1, where values close to ±1 indicate strong linear relationships. 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 a normal distribution, showing how many standard deviations an observation is from the mean. They are used to compare observed data against a theoretical normal model, facilitating tests for normality and correlation analysis.

Critical Value and Normality Tests

Critical values are threshold values from statistical tables used to decide whether to reject a null hypothesis. In normality tests, comparing a test statistic to a critical value helps determine if data significantly deviate from normality, guiding conclusions about the data distribution.