Question

Summary statistics are given for independent simple random samples from two populations. Use the nonpooled t-test to conduct the required hypothesis test. 1 = 75.1, s 1 = 4.5, n 1 = 11, 2 = 66.2, s 2 = 5.1, n 2 = 9 Perform a two-tailed hypothesis test using a significance level of α = 0.01.

Answer 1

The decision rule is

Reject the null hypothesis

Explanation:

From the question we are told that

The first sample mean is
`\= x_1 = 75.1`

The first sample standard deviation is
`s_1 = 4.5`

The first sample size
`n_1 = 11`

The second sample mean is
`\= x_2 = 66.2`

The second sample standard deviation is
`s_2 = 5.1`

The second sample size is
`n_2 = 9`

The significance level is
`\alpha = 0.01`

The null hypothesis is
`H_o : \mu_1 = \mu_2`

The alternative hypothesis is
`H_a : \mu_1 \\e \mu_2`

Generally the pooled standard deviation is mathematically represented as

`s = \sqrt{ ((n_ 1 - 1) s_1^2 + (n_2 - 1) s_2^2)/(n_1 + n_2 -2 ) }`

=>
`s = \sqrt{ ((11 - 1) 4.5^2 + (9 - 1) 5.1^2)/(11 + 9 -2 ) }`

=>
`s = 4.78`

Generally the degree of freedom for the is mathematically represented as

`df = [ ([(s_1^2 )/(n_1) + (s_2^2)/( n_2) ]^2)/([([(s_1^2)/(n_1) ]^2)/(n_1 - 1 ) ] + [([(s_2^2)/(n_2) ]^2)/(n_2 -1) ]) ]`

=>
`df = [ ([(4.5^2 )/(11) + (5.1^2)/(9)  ]^2)/([([(4.5^2)/(11) ]^2)/( 11 - 1 ) ] + [([(5.1^2)/(9) ]^2)/(9 -1)  ]) ]`

=>
`df = 16`

Generally the test statistics is mathematically represented as

`t = \frac{\= x_1 - \= x_2}{ \sqrt{(s_1^2 )/(n_1) + (s_2^2)/(n_2) } }`

=>
`t = \frac{ 75.1 - 66.2 }{ \sqrt{( 4.5^2 )/( 11) + (5.1^2)/( 9) } }`

=>
`t = 4.092`

Generally the student distribution table the probability of
`t = 4.092` to the right at a degree of freedom of
`df = 16` is

`P( T > 4.092 ) = 0.00042536`

Generally the p-value is mathematically represented as

`p-value = 2 * P(T > 4.092)`

=>
`p-value = 2 * 0.00042536`

=>
`p-value = 0.001`

From the value obtained we see that
`p-value < \alpha` hence we

The decision rule is

Reject the null hypothesis