Statistics 8.1, 8.2

What effect does increasing the sample size have on the​ probability? Provide an explanation for this result.

Increasing the sample size decreases

Is the statement below true or​ false? The mean of the sampling distribution of p hat is p.

The statement is true. The mean of the sample distribution of the sampling proportion equals the population​ proportion, p.

For the shape of the distribution of the sample proportion to be approximately​ normal, it is required that ​np (1 -p ) greater than or equals​______.

For the shape of the distribution of the sample proportion to be approximately​ normal, it is required that ​np (1 -p ) greater than or equals 10.

Why is the sampling distribution of x overbar approximately​ normal?

If a random variable X is normally​ distributed, the distribution of the sample​ mean, x over bar, is normally distributed. If the sample size is large​ enough, n greater than or equals 30, the sampling distribution is approximately normal regardless of the shape of the population. The sampling distribution of x overbar is approximately normal because the sample size is large enough.

Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why?

No because the Central Limit Theorem states that regardless of the shape of the underlying​ population, the sampling distribution of x overbar becomes approximately normal as the sample​ size, n, increases.

Without doing any​ computation, decide which has a higher​ probability, assuming each sample is from a population that is normally distributed with ri equals 100 and sigma equals 15. Explain your reasoning.

Both probabilities are based on a sample from the same population. The only difference is the sample size. The sampling distribution of the mean becomes narrower as the sample size is increased. Since the given​ interval, 90 less than or equal x less than or equals​110, contains the mean of​ 100, then more of the sampling distribution is contained in the interval as the sample size is increased. That means that the probability increases as the sample size increases.

Suppose the proportion of a population that has a certain characteristic is 0.45 . The mean of the sampling distribution of ModifyingAbove p hat from this population is ri p hat = ______.

Suppose the proportion of a population that has a certain characteristic is 0.45. The mean of the sampling distribution of p hat from this population is ri p hat = 0.45 .