Answered: The lengths of adult males' hands are…

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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Question

The lengths of adult males' hands are normally distributed with mean 185 mm and standard deviation is 7.4 mm. Suppose that 6 individuals are randomly chosen. Round all answers to 4 where possible.

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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Step 1

Given Information:

The lengths of adult males' hands are normally distributed Mean $μ = 185$ mm and Standard deviation $σ = 7.4$ mm.

Sample size (n) = 6

(a) If a sample of size $n$ is drawn from a normal population with mean $μ$ and standard deviation $σ$ , then the sampling distribution of sample mean $\bar{x}$ is approximately normally distributed with mean $μ_{\bar{x}} = μ$ and standard deviation $σ_{\bar{x}} = \frac{σ}{\sqrt{n}}$ .

So, mean of the sampling distribution of sample mean is $μ_{\bar{x}} = 185$

Standard deviation of the sampling distribution of sample mean is $σ_{\bar{x}} = \frac{7.4}{\sqrt{6}} = 3.021037349 \approx 3.02104$

Thus, $\bar{x} ~ N (185, 3.0210)$

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