A company in Maryland has developed a device that can be attached to car engines, which it believes will increase the miles per gallon that cars will get. The owners are interested in estimating the difference between mean mpg for cars using the device versus those that are not using the device. The following data represent the mpg for random independent samples of cars from each population. The variances are assumed equal and the populations normally distributed. With Device Without Device 22.6 26.9 23.4 24.4 28.4 20.8 29.0 20.8 29.3 20.2 20.0 26.0 ​ 28.1 ​ 25.6 Given this data, what is the observed value of the test statistic for the difference in mean mpg? About 0.72 About 2.18 About 25.45 mpg None of the above

Answer:

About 0.72

Step-by-step explanation:

Given the data:

With device : 22.6, 23.4, 28.4, 29, 29.3, 20.0

Without device :26.9, 24.4, 20.8, 20.8, 20.2, 26.0, 28.1, 25.6

Using calculator:

With device data:

Sample size (n1) = 6

Degree of freedom (df1) = 6 - 1 = 5

Mean(m1) = 25.45

Variance (s²1) = 15.631

Without device data:

Sample size (n2) = 8

Degree of freedom (df2) = n - 1 = 7

Mean (m2) = 24.1

Variance (s²2) = 9.54

s²t = ((df1/(df1 + df2)) * s²1) + ((df2/(df1 + df2)) * s²2)

s²t = ((5/(5+7)*15.63) + ((7/(5+7))*9.54)

= ((5/12) * 15.63) + ((7/12) * 9.54) = 12.0775

s²m1 = s²t/n1 = 12.0775/6 = 2.0123

s²m2 = s²t/n2 = 12.0775/8 = 1.511

T - statistic :

Tstat = (m1 - m2)/√(s²m1 + s²m2)

Tstat = (25.45 - 24.10) / √(2.0123 + 1.511)

Tstat= 1.350/√3.5233

Tstat = 1.350 / 1.8770455

Tstat = 0.7192153 = 0.72

In this exercise we want to use the knowledge of statistics and probability to calculate the statistical test that counts:

[tex]0.72[/tex]

First, we have to separate the data informed in the text and analyze it, like this:

Now doing with the device data calculations, we will find that:

Now doing without device data calculations, we will find that:

So to calculate the variance of the population we will use the data given above:

[tex]s~2t = ((df_1/(df_1 + df_2)) * s^2_1) + ((df_2/(df_1 + df_2)) * s^2_2)\\s^2t = ((5/(5+7)*15.63) + ((7/(5+7))*9.54)\\= ((5/12) * 15.63) + ((7/12) * 9.54) = 12.0775\\s^2m_1 = s^2t/n_1 = 12.0775/6 = 2.0123\\s^2m_2 = s^2t/n_2 = 12.0775/8 = 1.511[/tex]

For the calculation of the statistical test, we found that:

[tex]Tstat = (m_1 - m_2)/\sqrt{(s^2m_1 + s^2m_2)} \\Tstat = (25.45 - 24.10) / \sqrt{(2.0123 + 1.511)} \\Tstat= 1.350/\sqrt{3.5233} \\Tstat = 1.350 / 1.8770455\\Tstat = 0.7192153 = 0.72[/tex]

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