C955 Applied Probability and Statistics - Set 2 - Part 1

Correct Answer: Symmetric, bell-shaped, and defined by its mean and standard deviation

A normal distribution is symmetric and bell-shaped, completely determined by its mean, which locates the center, and its standard deviation, which sets the spread.

Correct Answer: 68%

The empirical rule states that roughly 68 percent of observations fall within one standard deviation, 95 percent within two, and 99.7 percent within three.

Correct Answer: 95%

By the empirical rule, about 95 percent of the values in a normal distribution fall within two standard deviations of the mean.

Correct Answer: 99.7%

About 99.7 percent of observations in a normal distribution fall within three standard deviations, making values beyond that range very rare.

Correct Answer: The number of standard deviations a value lies from the mean

A z-score standardizes a value by expressing its distance from the mean in standard deviation units, computed as (value minus mean) divided by standard deviation.

Correct Answer: 2

The z-score equals (75 minus 65) divided by 5, which is 10 divided by 5, giving 2.

Correct Answer: It lies 1.5 standard deviations below the mean

A negative z-score means the value is below the mean, and its magnitude gives the distance in standard deviations.

Correct Answer: Mean 0 and standard deviation 1

The standard normal distribution is the normal distribution with mean 0 and standard deviation 1, obtained by converting values to z-scores.

Correct Answer: 50%

Because the normal distribution is symmetric about its mean, exactly half of the distribution lies below the mean.

Correct Answer: 68%

The interval 60 to 80 is exactly one standard deviation on either side of the mean, which by the empirical rule contains about 68 percent of values.

Correct Answer: 95%

The interval 50 to 90 spans two standard deviations on each side of the mean, containing approximately 95 percent of the values.

Correct Answer: The distribution of a statistic across all possible samples of a given size

A sampling distribution describes how a statistic such as the sample mean varies from sample to sample, forming the basis for inference.

Correct Answer: The sampling distribution of the mean approaches normality as sample size increases

The Central Limit Theorem says that for sufficiently large samples, the sampling distribution of the mean is approximately normal regardless of the population's shape.

Correct Answer: It allows normal-based inference even when the population is not normal

Because the sampling distribution of the mean becomes approximately normal, standard normal-based confidence intervals and tests remain valid for many non-normal populations.

Correct Answer: The standard deviation of the sampling distribution of the mean

The standard error measures how much sample means vary around the population mean, computed as the population standard deviation divided by the square root of n.

Correct Answer: It decreases

Because the standard error divides by the square root of the sample size, larger samples produce smaller standard errors and therefore more precise estimates.

Correct Answer: It must be multiplied by four

Standard error is inversely proportional to the square root of n, so reducing it by half requires quadrupling the sample size.

Correct Answer: A single value used to estimate a population parameter

A point estimate is a single number, such as a sample mean, used as the best available estimate of an unknown population parameter.

Correct Answer: A range of plausible values for a parameter with a stated confidence level

A confidence interval gives a range of plausible parameter values, constructed so that the stated percentage of such intervals would contain the true parameter.

Correct Answer: About 95% of intervals built this way would contain the true parameter

Confidence refers to the long-run performance of the procedure: 95 percent of intervals constructed this way capture the parameter, though any single interval either does or does not.