8.1-8.2
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 nincreases, 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
