you wish to test the following claim ($h_a$) at a significance level of $alpha = 0.01$. $h_0:mu_1=mu_2$…

you wish to test the following claim ($h_a$) at a significance level of $alpha = 0.01$. $h_0:mu_1=mu_2$ $h_a:mu_1<mu_2$ you believe both populations are normally distributed, but you do not know the standard deviations for either, and you have no reason to believe the variances of the two populations are equal. you obtain a sample of size $n_1 = 12$ with a mean of $m_1 = 54.5$ and a standard deviation of $sd_1 = 20.3$ from the first population. you obtain a sample of size $n_2 = 13$ with a mean of $m_2 = 75.3$ and a standard deviation of $sd_2 = 11.7$ from the second population. what is the test statistic for this sample? (report answer accurate to two decimal places.) test statistic = what is the p - value for this sample? (report answer accurate to four decimal places.) p - value = the p - value is... less than (or equal to) $alpha$ greater than $alpha$ this test statistic leads to a decision to... reject the null accept the null fail to reject the null

you wish to test the following claim ($h_a$) at a significance level of $alpha = 0.01$. $h_0:mu_1=mu_2$ $h_a:mu_1<mu_2$ you believe both populations are normally distributed, but you do not know the standard deviations for either, and you have no reason to believe the variances of the two populations are equal. you obtain a sample of size $n_1 = 12$ with a mean of $m_1 = 54.5$ and a standard deviation of $sd_1 = 20.3$ from the first population. you obtain a sample of size $n_2 = 13$ with a mean of $m_2 = 75.3$ and a standard deviation of $sd_2 = 11.7$ from the second population. what is the test statistic for this sample? (report answer accurate to two decimal places.) test statistic = what is the p - value for this sample? (report answer accurate to four decimal places.) p - value = the p - value is... less than (or equal to) $alpha$ greater than $alpha$ this test statistic leads to a decision to... reject the null accept the null fail to reject the null

Answer

Explanation:

Step1: Calculate the test - statistic formula

The test - statistic for two independent samples with unknown and unequal variances is given by the Welch's t - test formula: [t=\frac{(M_1 - M_2)-(\mu_1-\mu_2)}{\sqrt{\frac{SD_1^{2}}{n_1}+\frac{SD_2^{2}}{n_2}}}] Under the null hypothesis (H_0:\mu_1=\mu_2), so ((\mu_1 - \mu_2) = 0). We are given (n_1 = 12), (M_1=54.5), (SD_1 = 20.3), (n_2 = 13), (M_2 = 75.3), and (SD_2=11.7). [t=\frac{(54.5 - 75.3)-0}{\sqrt{\frac{20.3^{2}}{12}+\frac{11.7^{2}}{13}}}]

Step2: Calculate the denominator

First, calculate (\frac{20.3^{2}}{12}+\frac{11.7^{2}}{13}): (\frac{20.3^{2}}{12}=\frac{412.09}{12}\approx34.3408) (\frac{11.7^{2}}{13}=\frac{136.89}{13}\approx10.53) (\frac{20.3^{2}}{12}+\frac{11.7^{2}}{13}\approx34.3408 + 10.53=44.8708) (\sqrt{\frac{20.3^{2}}{12}+\frac{11.7^{2}}{13}}\approx\sqrt{44.8708}\approx6.70)

Step3: Calculate the test - statistic

[t=\frac{54.5 - 75.3}{6.70}=\frac{- 20.8}{6.70}\approx - 3.10]

Step4: Calculate the degrees of freedom

The degrees of freedom for Welch's t - test is calculated using the formula: [df=\frac{\left(\frac{SD_1^{2}}{n_1}+\frac{SD_2^{2}}{n_2}\right)^{2}}{\frac{(SD_1^{2}/n_1)^{2}}{n_1 - 1}+\frac{(SD_2^{2}/n_2)^{2}}{n_2 - 1}}] (\frac{SD_1^{2}}{n_1}=\frac{412.09}{12}\approx34.3408), (\frac{SD_2^{2}}{n_2}=\frac{136.89}{13}\approx10.53) (\left(\frac{SD_1^{2}}{n_1}+\frac{SD_2^{2}}{n_2}\right)^{2}=(34.3408 + 10.53)^{2}=44.8708^{2}\approx2013.3) (\frac{(SD_1^{2}/n_1)^{2}}{n_1 - 1}=\frac{34.3408^{2}}{11}=\frac{1179.28}{11}\approx107.21) (\frac{(SD_2^{2}/n_2)^{2}}{n_2 - 1}=\frac{10.53^{2}}{12}=\frac{110.88}{12}\approx9.24) [df=\frac{2013.3}{107.21+9.24}=\frac{2013.3}{116.45}\approx17.29\approx17]

Step5: Calculate the p - value

Since this is a one - tailed test ((H_a:\mu_1<\mu_2)) and (t\approx - 3.10) with (df = 17), using a t - distribution table or a calculator (e.g., in R: pt(-3.1,17)), the p - value is (P(t_{17}<-3.10)\approx0.0049)

Step6: Make a decision

Since the significance level (\alpha = 0.01) and the p - value (0.0049<0.01), we reject the null hypothesis.

Answer:

test statistic = - 3.10 p - value = 0.0049 The p - value is less than (or equal to) (\alpha) This test statistic leads to a decision to reject the null