Answer to Question #127383 in Statistics and Probability for Yeap Kai Wei

Question

Question 3
An automobile manufacturer collects mileage data for a sample of 10 cars in various weight categories by use of a standard grade of gasoline with and without a particular additive. The Engines were tuned to the same specifications before each run, and the same drivers were used for the two gasoline conditions (with the driver in fact being unaware of which gasoline was being used on a particular run). The mileage data is given in the table below:

Automobile 1 2 3 4 5 6 7 8 9 10
Mileage with Additive 36.5 36.1 32.0 29.3 28.4 25.7 24.2 22.6 21.0 20.3
Mileage without Additive 36.2 35.9 32.3 29.6 28.1 25.5 23.9 22.0 21.4 20.0

(a) test whether the additive is effective in increasing the mileage at 1% level of significance.
(12 marks)
(b) Do you think it is necessary to use the same drivers for two gasoline conditions? Explain your reasoning.
(3 marks)

Answer 1

a)The sampling method for each sample is simple random sampling. The test is conducted on paired data (same sample with and without additive). Hence we use paired t test here.

state hypothesis:

H₀:$\mu _{d}=0$

H_a:$\mu _{d}>0$

Automobile 1 2 3 4 5 6 7 8 9 10

with Additive 36.5 36.1 32.0 29.3 28.4 25.7 24.2 22.6

21.0 20.3

without Additive 36.2 35.9 32.3 29.6 28.1 25.5 23.9 22.0 21.4 20.0

difference 0.3 0.2 -0.3 -0.3 0.3 0.2 0.3 0.6

-0.4 0.3

Using ti84 , sample mean = $\overline{d}$ =0.12, $s_{\overline{d}}=0.333$ , n= 10

The t-statistic is computed as shown in the following formula:

t= $\frac{\overline{d}}{\frac{s_{\overline{d}}}{\sqrt{n}}}$

= $\frac{0.12}{\frac{0.333}{\sqrt{10}}}$

=1.141

using t table ,using the P-value approach: The p-value is p = 0.1417

since p=0.1417≥0.01, it is concluded that the null hypothesis is not rejected.

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1

>  μ2​, at the 0.01 significance level.

b)yes it is necessary to use the same drivers for two gasoline conditions since we have same sample for both with additive and non additive.