Question

Read the following scenario and complete each of the seven problems below:

A new car manufacturing company has emerged and has claimed that its new hybrid car, the Pusho, gets a better gas mileage than the highest ranked Toyota Prius. Consumer Reports Magazine decides to test this claim at a 5% level of significance. Consumer Reports randomly selects 10 of each type of car, calculates the miles per gallon for each car in the study, and records the data in the table below. Assume miles per gallon of the cars is normally distributed.

Pusho 54.1     52.4    55.7 49.7   50.6      48.9    51.8    54.5    56.9    49.8

Prius   53.2     54.3    49.8    50.1    50.5     56.1    47.8    53.4    56.8    48.7

A. Evaluate the claim that the Pusho gets a better gas mileage than the highest ranked Prius using the data from the Consumer Reports study.

1. Identify the type of test you will use to test this claim. Explain your reasoning. 2. State the null and alternate hypotheses. 3. Conduct the hypothesis test and determine the p-value. 4. State your conclusion about the claim. 5. Construct a 90% confidence interval for this study.

1.

2.

3.

4.

5.

Suppose that Toyota makes a counter-claim that their Prius has a higher gas mileage than Pusho. How could the alternative hypotheses from Part A be changed to test Toyota’s claim? Conduct the hypothesis test at the 5% level of significance for Toyota’s claim using the data above and determine the p-value. State your conclusion about the claim. Construct a 98% confidence interval for this study.

1.

2.

3.

4.

C. Based on your analysis of both claims from the makers of Pusho and Prius, what statement can be made about the miles per gallon of the two cars? Explain your reasoning.

Answer 1

For pusho

= 52.44, s1 = 2.614, n1 = 10

For Prius

= 52.07, s2 = 3.122, n2 = 10

A)1) We will use independent two-sample t-test.

2) H₀:

    H₁:

3) The test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                             = (52.44 - 52.07)/sqrt((2.614)^2/10 + (3.122)^2/10)

                            = 0.287

df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

    = ((2.614)^2/10 + (3.122)^2/10)^2/(((2.614)^2/10)^2/9 + ((3.122)^2/10)^2/9)

   = 17

P-value = P(T > 0.287)

             = 1 - P(T < 0.287)

             = 1 - 0.6112

            = 0.3888

4) Since the P-value is greater than the significance level (0.3888 > 0.05), so we should not reject the null hypothesis.

So at 5% significance level there is not sufficient evidence to support the claim that the Pusho gets a better gas mileage than the highest ranked Prius.

5) At 90% confidence interval the critical value is t^* = 1.740

The 90% confidence interval is

() +/- t^* * sqrt(s1^2/n1 + s2^2/n2)

= (52.44 - 52.07) +/- 1.74 * sqrt((2.614)^2/10 + (3.122)^2/10)

= 0.37 +/- 2.24

= -1.87, 2.61

B) 1) We will use independent two-sample t-test.

2) H₀:

      H₁:

3) The test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                             = (52.44 - 52.07)/sqrt((2.614)^2/10 + (3.122)^2/10)

                            = 0.287

df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

    = ((2.614)^2/10 + (3.122)^2/10)^2/(((2.614)^2/10)^2/9 + ((3.122)^2/10)^2/9)

   = 17

P-value = P(T < 0.287)

             = 0.6112

4) Since the P-value is greater than the significance level (0.6112 > 0.05), so we should not reject the null hypothesis.

So at 5% significance level there is not sufficient evidence to support the claim that Prius has a higher gas mileage than Pusho.

5) At 98% confidence interval the critical value is t^* = 2.567

The 98% confidence interval is

() +/- t^* * sqrt(s1^2/n1 + s2^2/n2)

= (52.44 - 52.07) +/- 2.567 * sqrt((2.614)^2/10 + (3.122)^2/10)

= 0.37 +/- 3.305

= -2.935, 3.657

C) Since in both the claims, we concluded that the null hypothesis will not be rejected. So we can conclude that the miles per gallon of the two cars are same.