variable of a population has a mean of mu= 89 and a standard deviation of sigma
ID: 3127216 • Letter: V
Question
variable of a population has a mean of mu= 89 and a standard deviation of sigma = 27 Identify the sampling distribution of the sample mean for samples of size 81. In answering part (a), what assumptions did you make about the distribution of the variable? Can you answer part (a) if the sample size is 16 instead of 81? Why or why not? What is the shape of the sampling distribution? bimodal uniform skewed normal What is the mean of the sampling distribution? Mu_x^-= (Simplify your answer.) What is the standard deviation of the sampling distribution sigma_x^-= (Simplify your answer.) In answering part (a), what assumptions did you make about the distribution of the variable? it was assumed that the variable was normally distributed. No assumptions were made because, for a relatively large sample size, the sampling distribution is normal, regardless of the distribution of the variable under consideration. It was assumed that the distribution of the variable was unimodal. No assumptions were made because the sampling distribution is normal, regardless of the distribution of the variable under consideration. Can you answer part (a) if the sample size is 16 instead of 81? Why or why not? No, because the sample size needs to be at least 30 if the distribution of the variable is unknown. Yes, because the sample size makes no difference in determining the sampling distribution. No, because the sample size must be less than 5 if the distribution of the variable is unknown. Yes, because the sample size must be less than 30 if the distribution of the variable is unknown.Explanation / Answer
(a)
as sample size=n=81, as n>30
the sample is large sample and distribution is normal
Properties of normal distribution:
Every normal distribution has certain properties. You use these properties to determine the relative standing of any particular result on the distribution, and to find probabilities. The properties of any normal distribution are as follows:
Its shape is symmetric (that is, when you cut it in half the two pieces are mirror images of each other).
Its distribution has a bump in the middle, with tails going down and out to the left and right.
The mean and the median are the same and lie directly in the middle of the distribution (due to symmetry).
Its standard deviation measures the distance on the distribution from the mean to the inflection point (the place where the curve changes from an "upside-down-bowl" shape to a "right-side-up-bowl" shape).
Because of its unique bell shape, probabilities for the normal distribution follow the Empirical Rule, which says the following:
(a)
shape of sampling distribution is normal
About 68 percent of its values lie within one standard deviation of the mean. To find this range, take the value of the standard deviation, then find the mean plus this amount, and the mean minus this amount.
About 95 percent of its values lie within two standard deviations of the mean. (Here you take 2 times the standard deviation, then add it to and subtract it from the mean.)
Almost all of its values (about 99.7 percent of them) lie within three standard deviations of the mean. (Take 3 times the standard deviation and add it to and subtract it from the mean.)
The sampling distribution of the sample mean for samples of size 81 is approximately normally distributed with mean 89 and standard deviation 27/sqrt(81) = 27/9 = 3
mean of sampling distribution=89
standard deviatioon of sampling distribution=3
(b)A
it was assumed that the variable was normally distributed.
c)A
if sample size is 16 ,sampling distribution does not fall into normal distrbution.
For a distribution to be normal n>=30,if the distibution of the variable is unknown
with n=81 ,it satisfies the distribution to be normal.
Mean of sample= µ, the standard deviation = 2 /n, and the sampling distribution of is normally distributed if the population is normally distributed.
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