C955 Applied Probability and Statistics - Set 5 - Part 1

Correct Answer: 30

The values sum to 150, and 150 divided by 5 equals 30.

Correct Answer: 30

With five ordered values, the median is the middle one, which is 30. Here the mean and median coincide because the data are symmetric.

Correct Answer: 40

The range is the maximum minus the minimum: 50 minus 10 equals 40.

Correct Answer: Both 2 and 6

Both 2 and 6 appear twice, more than 4, so the data set is bimodal with modes 2 and 6.

Correct Answer: It increases by 5

Adding a constant to every value shifts the entire distribution, so the mean increases by that same constant.

Correct Answer: It stays the same

Adding a constant shifts all values equally without changing how spread out they are, so the standard deviation is unchanged.

Correct Answer: It is multiplied by 3

Multiplying every value by a constant scales the spread by the same factor, so the standard deviation is multiplied by 3.

Correct Answer: 50%

Because the normal distribution is symmetric about the mean, exactly half of the area lies above it.

Correct Answer: The mean

A z-score of 0 indicates a value exactly at the mean, since the score measures distance from the mean in standard deviation units.

Correct Answer: The student with z = 1.2

Z-scores standardize performance relative to each exam's mean and spread, so the higher z-score of 1.2 indicates the relatively better performance.

Correct Answer: Allowing comparison of values from different distributions

Converting to z-scores expresses values on a common scale relative to each distribution's mean and spread, enabling meaningful comparison across different distributions.

Correct Answer: 30

A z-score of -2 lies two standard deviations below the mean: 50 minus 2 times 10 equals 30.

Correct Answer: About 68%

By the empirical rule, about 68 percent of the standard normal distribution lies within one standard deviation of the mean, here between -1 and 1.

Correct Answer: 2

The standard error equals the standard deviation divided by the square root of the sample size: 10 divided by the square root of 25 (which is 5) equals 2.

Correct Answer: It is halved

Quadrupling the sample size doubles the square root of n from 5 to 10, so the standard error is halved from 2 to 1.

Correct Answer: Less precision

A wider interval reflects greater uncertainty and therefore less precision, often resulting from a smaller sample or higher variability.

Correct Answer: A larger sample size

A larger sample reduces the standard error, yielding a narrower interval. Higher confidence and greater variability both widen the interval.

Correct Answer: 50

The point estimate is the center of the interval, which is the average of the endpoints: (45 plus 55) divided by 2 equals 50.

Correct Answer: 5

The margin of error is half the interval width, or the distance from the center to an endpoint: 55 minus 50 equals 5.

Correct Answer: Reject the null hypothesis

Because the p-value 0.03 is less than the significance level 0.05, the result is statistically significant and the null hypothesis is rejected.