Respuesta :
Answer:
a) At a significance level of 0.1, there is not enough evidence to support the claim that the mean secret ingredient percentage in Big Micks from the two stores are not equal.
b) The 90% confidence interval for the difference between means is (-0.632, 0.022).
As the value 0 is included as a probable value for the difference of means, we can conclude that there is no enough evidence that the difference of means is significantly different from 0.
Both the hypothesis test and the confidence interval have the same result, as both use the same sample and assumptions.
Step-by-step explanation:
a) This is a hypothesis test for the difference between populations means.
The claim is that the mean secret ingredient percentage in Big Micks from the two stores are not equal.
Then, the null and alternative hypothesis are:
[tex]H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2\neq 0[/tex]
The significance level is 0.1.
The sample 1 (corner stores), of size n1=53 has a mean of 0.035 and a standard deviation of 1.046.
The sample 2 (main stores), of size n2=51 has a mean of 0.34 and a standard deviation of 0.96.
The difference between sample means is Md=-0.305.
[tex]M_d=M_1-M_2=0.035-0.34=-0.305[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{1.046^2}{53}+\dfrac{0.96^2}{51}}\\\\\\s_{M_d}=\sqrt{0.021+0.018}=\sqrt{0.039}=0.1968[/tex]
Then, we can calculate the t-statistic as:
[tex]t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{-0.305-0}{0.1968}=\dfrac{-0.305}{0.1968}=-1.55[/tex]
The degrees of freedom for this test are:
[tex]df=n_1+n_2-2=53+51-2=102[/tex]
This test is a two-tailed test, with 102 degrees of freedom and t=-1.55, so the P-value for this test is calculated as (using a t-table):
[tex]\text{P-value}=2\cdot P(t<-1.55)=0.12[/tex]
As the P-value (0.12) is greater than the significance level (0.1), the effect is not significant.
The null hypothesis failed to be rejected.
At a significance level of 0.1, there is not enough evidence to support the claim that the mean secret ingredient percentage in Big Micks from the two stores are not equal.
b) We have to calculate a 90% confidence interval for the difference between means.
The critical t-value for a 90% confidence interval is t=1.66.
The margin of error (MOE) can be calculated as:
[tex]MOE=t\cdot s_{M_d}=1.66 \cdot 0.1968=0.327[/tex]
Then, the lower and upper bounds of the confidence interval are:
[tex]LL=M_d-t \cdot s_{M_d} = -0.305-0.327=-0.632\\\\UL=M_d+t \cdot s_{M_d} = -0.305+0.327=0.022[/tex]
The 90% confidence interval for the difference between means is (-0.632, 0.022).
As the value 0 is included as a probable value for the difference of means, we can conclude that there is no enough evidence that the difference of means is significantly different from 0.
