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Name: Two samples are taken with the following sample means, sizes, and stan
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Description: Two samples are taken with the following sample means, sizes, and standard deviations ¯x1x¯1 = 39 ¯x2x¯2 = 35 n1n1 = 74 n2n2 = 63 s1s1 = 4 s2s2 = 5 Find a 86% confidence interval, round answers to the nearest hundredth. ____________ < μ1−μ2μ1-μ2 <___

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Two samples are taken with the following sample means, sizes, and standard deviations ¯x1x¯1 = 39 ¯x2x¯2 = 35 n1n1 = 74 n2n2 = 63 s1s1 = 4 s2s2 = 5 Find a 86% confidence interval, round answers to the nearest hundredth. ____________ < μ1−μ2μ1-μ2 <________________

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

We need to construct the $86 %$ confidence interval for the difference between the population means $μ_{1} - μ_{2}$ , for the case that the population standard deviations are not known. The following information has been provided about each of the samples:

Sample Mean 1 $(\bar X_1)$ =	$39$
Sample Standard Deviation 1 $(s_1)$ =	$4$
Sample Size 1 $(N_1)$ =	$74$
Sample Mean 2 $(\bar X_2)$ =	$35$
Sample Standard Deviation 2 $(s_2)$ =	$5$
Sample Size 2 $(N_2)$ =	$63$

Based on the information provided, we assume that the population variances are equal, so then the number of degrees of freedom are $df = n_1 + n_2 -2 = 74 + 63 - 2 = 135$

$= 74 + 63 - 2$

$df = 135$ .

The critical value for $α = 0.14$ and $d f = 135$ degrees of freedom is $t_{c} = t_{1 - α /2; n - 1} = 1.485$ .

$\begin{array}{ccl} s_p & = & \displaystyle \sqrt{ \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2} } \\ \\ & = & \displaystyle \sqrt{ \frac{(74-1)\times 4^2 + (63-1) \times 5^2}{74 + 63 - 2} } \\ \\ & = & 4.487 \end{array}$

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