Statistics

Step 1: Calculate the original sum of the 10 items.

Step 2: Calculate the total change made to the sum. - For the first seven items, 5 is added to each. Total increase = . - For the last three items, 3 is subtracted from each. Total decrease = . - The net change in the sum is .

Step 3: Calculate the new sum.

Step 4: Calculate the new mean. The number of items remains 10.

Step 1: Observe the relationship between X and Y. From the table, it is clear that for every value of X, the corresponding value of Y is . This is a perfect, but non-linear (quadratic), relationship.

Step 2: Understand what the coefficient of correlation (Pearson's r) measures. The Pearson correlation coefficient measures the strength and direction of the linear relationship between two variables.

Step 3: Calculate the means of X and Y.

Step 4: Calculate the covariance between X and Y. The formula for covariance is . Since the sum is 0, the covariance is 0.

Step 5: Determine the correlation coefficient. The correlation coefficient is . Since the numerator (covariance) is 0, the correlation coefficient is 0, provided the standard deviations are not zero (which they are not). A value of 0 indicates no linear relationship.

Step 1: Recall the definition of variance.
. Let .

Step 2: Apply the definition to .
. Let and .

Step 3: Expand the squared term.

Step 4: Use the linearity of expectation.

Step 5: Recognize the definitions of variance and covariance. Substituting these back gives the final formula:

Step 1: Define the parameters and hypotheses. This is a two-sample test for proportions. Let and be the true error proportions for Team A and Team B. - Sample proportion for A: with . - Sample proportion for B: with . The hypothesis is to test if Team A performed better, meaning they have a lower error rate. So, and .

Step 2: Calculate the pooled proportion ( ). Under , we assume the proportions are equal. We estimate this common proportion by pooling the data.

Step 3: State the formula for the test statistic (Z).

Step 4: Calculate the values and the final test statistic. - Numerator: . - Pooled proportion: . So, . - Denominator: - Test Statistic Z: