Question

A random sample of n1 = 10 winter days in Denver gave a sample mean pollution index x1 = 43. Previous studies show that σ1 = 21. For Englewood (a suburb of Denver), a random sample of n2 = 12 winter days gave a sample mean pollution index of x2 = 36. Previous studies show that σ2 = 13.

Assume the pollution index is normally distributed in both Englewood and Denver.

(a) Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance.

(i) What is the level of significance? State the null and alternate hypotheses.

H0: μ1 = μ2; H1: μ1 > μ2

H0: μ1 = μ2; H1: μ1 < μ2

H0: μ1 < μ2; H1: μ1 = μ2

H0: μ1 = μ2; H1: μ1 ≠ μ2

() What sampling distribution will you use? What assumptions are you making?

The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

The Student's t. We assume that both population distributions are approximately normal with known standard deviations.

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate. (Test the difference μ1 − μ2. Round your answer to two decimal places.) (iii) Find (or estimate) the P-value. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value.

(iv) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

() Interpret your conclusion in the context of the application.

Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

(b) Find a 99% confidence interval for μ1 − μ2. (Round your answers to two decimal places.) lower limit

upper limit

Explain the meaning of the confidence interval in the context of the problem.

Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.

Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.

Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.

Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver

Answer 1

H0: μ1 = μ2; H1: μ1 ≠ μ2

...........

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

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sample #1   ------->              
mean of sample 1,    x̅1=   43          
population std dev of sample 1,   σ1 =    21          
size of sample 1,    n1=   10          
                  
sample #2   --------->              
mean of sample 2,    x̅2=   36          
population std dev of sample 2,   σ2 =    13          
size of sample 2,    n2=   12          
                  
difference in sample means = x̅1 - x̅2 =    43   -   36   =   7
                  
std error , SE =    √(σ1²/n1+σ2²/n2) =    7.6278          
                  
Z-statistic = ((x̅1 - x̅2)-µd)/SE =    7   /   7.6278   =   0.92
                        
                  
p-value =        0.3588   [excel formula =2*NORMSDIST(z)]      
Desison:   p-value>α , Do not reject null hypothesis              

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

............

Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

.............

Level of Significance ,    α =    0.01          
z-critical value =    Z α/2 =    2.5758   [excel function =normsinv(α/2) ]      
                  
std error , SE =    √(σ1²/n1+σ2²/n2) =    7.6278          
margin of error, E = Z*SE =    2.5758   *   7.628   =   19.6479
                  
difference of means = x̅1 - x̅2 =    43   -   36   =   7.000

confidence interval is                   
Interval Lower Limit= (x̅1 - x̅2) - E =    7.000   -   19.648   =   -12.65
Interval Upper Limit= (x̅1 - x̅2) + E =    7.000   +   19.648   =   26.65

...............

Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.

..............

THANKS

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