Answered: A variable of a population has a mean…

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.4: Distributions Of Data

Problem 19PFA

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A variable of a population has a mean of μ=86 and a standard deviation of σ=7.

a. Identify the sampling distribution of the sample mean for samples of size 49.

b. In answering part (a), what assumptions did you make about the distribution of the variable?

c. Can you answer part (a) if the sample size is 16 instead of 49? Why or why not?

a. What is the shape of the sampling distribution?

bimodal

normal

skewed

uniform

What is the mean of the sampling distribution?

μx=_________________

(Simplify your answer.)

What is the standard deviation of the sampling distribution?

σx=____________________

(Simplify your answer)

b. In answering part (a), what assumptions did you make about the distribution of the variable?

A. It was assumed that the distribution of the variable was unimodal.

B. 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.

C. It was assumed that the variable was normally distributed.

D. No assumptions were made because the sampling distribution is normal, regardless of the distribution of the variable under consideration.

c. Can you answer part (a) if the sample size is 16 instead of 49?
Why or why not?

A. No, because the sample size needs to be at least 30 if the distribution of the variable is unknown.

B. No, because the sample size must be less than 5 if the distribution of the variable is unknown.

C. Yes, because the sample size must be less than 30 if the distribution of the variable is unknown.

D. Yes, because the sample size makes no difference in determining the sampling distribution.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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