Respuesta :
Answer:
a
The null hypothesis is [tex]H_o : \mu_1 = \mu_2[/tex]
The alternative hypothesis [tex]H_a : \mu_1 > \mu_2[/tex]
b
[tex]p-value = 0.232[/tex]
c
The decision rule is
Fail to reject the null hypothesis
Step-by-step explanation:
From the question we are told that
The value given is
S/N
1 7 5
2 4 3
3 8 7
4 8 8
5 7 9
6 7 5
7 6 5
Generally the sample mean for the first sample is mathematically represented as
[tex]\= x _1 = \frac{\sum x_i }{n}[/tex]
=> [tex]\= x _1 = \frac{7 +4 + \cdots + 6}{7}[/tex]
=> [tex]\= x _1 = 6.714[/tex]
Generally the sample mean for the second sample is mathematically represented as
[tex]\= x _2 = \frac{\sum x_i }{n}[/tex]
=> [tex]\= x _2 = \frac{5 + 3+ \cdots + 5}{7}[/tex]
=> [tex]\= x _2 = 6[/tex]
Generally the sample standard deviation for the first sample is mathematically represented as
[tex]s_1 = \sqrt{\frac{\sum (x_i - \= x_1)^2 }{n-1 } }[/tex]
=> [tex]s_1 = \sqrt{\frac{ (7 - 6.714 )^2 +(4 - 6.714 )^2 + \cdots + (6 - 6.714 )^2 }{7-1 } }[/tex]
=> [tex]s_1 = 1.905[/tex]
Generally the sample standard deviation for the second sample is mathematically represented as
[tex]s_2 = \sqrt{\frac{\sum (x_i - \= x_2)^2 }{n-1 } }[/tex]
=> [tex]s_2 = \sqrt{\frac{ (5 - 6.714 )^2 +(3 - 6.714 )^2 + \cdots + (5 - 6.714 )^2 }{7-1 } }[/tex]
=> [tex]s_1 = 4.33[/tex]
Generally the pooled standard deviation is
[tex]s = \sqrt{\frac{(n_1 - 1 )s_1^2 + (n_2 - 1 )s_2^2}{n_1 + n_2 -2 } }[/tex]
=> [tex]s = \sqrt{\frac{(7 - 1 )1.905^2 + (7 - 1 )4.333^2}{7 + 7 -2 } }[/tex]
=> [tex]s = 1.766[/tex]
The null hypothesis is [tex]H_o : \mu_1 = \mu_2[/tex]
The alternative hypothesis [tex]H_a : \mu_1 > \mu_2[/tex]
Generally the test statistics is mathematically represented as
[tex]t = \frac{\= x _1 - \= x_2 }{s * \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} }[/tex]
=> [tex]t = \frac{6.714 - 6 }{1.766 * \sqrt{\frac{1}{7} + \frac{1}{7}} }[/tex]
=> [tex]t = 0.757[/tex]
Generally the degree of freedom is mathematically represented as
[tex]df = n_1 + n_2 - 2[/tex]
=> [tex]df = 7 + 7 - 2[/tex]
=> [tex]df = 12[/tex]
From the t distribution table the probability of [tex]t = 0.757[/tex] at a degree of freedom of [tex]df = 12[/tex] is
[tex]t_{ 0.757 , 12} = 0.232[/tex]
Generally the p-value is
[tex]p-value = t_{ 0.757 , 12} = 0.232[/tex]
From the values obtained we see that [tex]p-value > \alpha[/tex] hence
The decision rule is
Fail to reject the null hypothesis
