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Since the population standard deviations are not known and the sample standard deviations are given, at distribution will be used. Recall that the t distribution is a symmetric bell-shaped curve that has unique curves for each degree of freedom. The degrees of freedom is calculated as follows where s, is the standard deviation of the sample taken from population 1, n, is the size of the sample taken from population 1, s, is the standard deviation of the sample taken from population 2, n, is the size of the sample taken from population 2. 2 52 + n2 df = 2 ni - 1 1 + n2 - 1 Recall the given information. Sample 1 Sample 2 ni = 40 n2 = 50 = 32.2 X2 = 30.1 $1 = 2.8 $2 = 4.1 Use these values to find the degrees of freedom, rounding the result to one decimal place. 2 2 $2 2 -1 + df 2 2 + 52 - 1 2.8 40 + n2 4.1 22 50 2 + 1 4.1 50 40 - 1 40 x Since the t distribution table uses only integer valued degrees of freedom, this value should be an integer. In the interest of being conservative, the degrees of freedom value should be rounded down to the nearest integer. Therefore, the degrees of freedom used is X.

Sagot :

The degrees of freedom used in the the t distribution is 86.

We have to calculate the degrees of freedom using the given formula. The t distribution is used as the population mean and standard deviation is not known. The data of two samples from two different population are given, using which we have to compute degrees of freedom.

Given,

n1=40, n2=50

[tex]\bar{x_{1} }[/tex]=32.2 , [tex]\bar{x_{2} }[/tex]= 30.1

s1=2.8 , s2= 4.1

The

The degrees of freedom formula is given by

[tex]df=\frac{\left(\frac{s_1{ }^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{1}{n_1-1}\left(\frac{s_1^2}{n_1}\right)^2+\frac{1}{n_2-1}\left(\frac{s_2^2}{n_2}\right)^2} \\[/tex]

Substituting the given values, we get

[tex]df=\frac{\left(\frac{2.8^2}{40}+\frac{4.1^2}{50}\right)^2}{\frac{1}{40-1}\left(\frac{2.8^2}{40}\right)^2+\frac{1}{50-1}\left(\frac{4.1^2}{50}\right)^2}[/tex]

Further simplifying, we get

[tex]d f=\frac{(0.5322)^2}{0.0032917}[/tex]

df= 86.04

Therefore, the degrees of freedom used after rounding off to the nearest integer is 86.

To know more about degrees of freedom here:

https://brainly.com/question/16972231#

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