8.1-8.2

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Suppose a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x has mean μx=​______and standard deviation σx=​______.

"μ", "σ overbar square root of n"

true or false. To cut the standard error of the mean in​ half, the sample size must be doubled.

False. The sample size must be increased by a factor of four to cut the standard error in half.

Suppose Jack and Diane are each attempting to use a simulation to describe the sampling distribution from a population that is skewed right with mean 70 and standard deviation 15. Jack obtains 1000 random samples of size n=5 from the​ population, finds the mean of the​ means, and determines the standard deviation of the means. Diane does the same​ simulation, but obtains 1000 random samples of size n=40 from the population. Complete parts​ (a) through​ (c) below. Describe the shape you expect for Jack​'s distribution of sample means. Describe the shape you expect for Diane​'s distribution of sample means

Jack​'s distribution is expected to be skewed right​, but not as much as the original distribution. Diane​'s distribution is expected to be approximately normal

A simple random sample of size n=40 is obtained from a population with μ=77 and σ=5. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally​ distributed? Why? What is the sampling distribution of x​? Does the population need to be normally distributed for the sampling distribution of x 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 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 μ=100 and σ=15. Explain your reasoning. ​(a)​ P(90≤x≤​110) for a random sample of size n=10 ​(b)​ P(90≤x≤​110) for a random sample of size n=20

P(90≤x≤​110) for a random sample of size n=20 has a higher probability. As n​increases, the standard deviation decreases.

True or False​: The population proportion and sample proportion always have the same value.

false

The​ _____ _____, denoted p​, is given by the formula p=​_____, where x is the number of individuals with a specified characteristic in a sample of n individuals.

sample proportion, x/n​

The standard deviation of the sampling distribution of x​, denoted σx​, is called the​ _____ _____ of the​ _____.

standard, error, mean

true or​ false? The distribution of the sample​ mean, x​, will be normally distributed if the sample is obtained from a population that is normally​ distributed, regardless of the sample size.

true

true or​ false? The mean of the sampling distribution of p is p.

true


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