Q. 9.118 - Elementary Statistics

Question

Dirt Bikes. Dirt bikes are simpler and lighter motorcysles that are designed for off-road events. Specifications for dirt bikes can be found through Motorcycle USA on their website www.motorcycle-usa.com. A random sample of 30 dirt bikes have a mean fuel capacity of $1.91$ gallons with a standard deviation of 0.74 gallons. At the $10\%$ significance level, do the data provide sufficient evidence to conclude that the mean fuel tank capacity of all dirt bikes is less than 2 gallons?

Step-by-Step Solution

Answer

Critical value: From the Excel add-in (PHStat) output, the critical value is -1.3114.

P-value: From the Excel add-In (PHStat)output, the P-value is 0.2553 .

Step 1: Given Information

Examine if the data supports the conclusion that the average fuel tank capacity of all dirt motorcycles is less than 2 litres.

State the null and alternative hypothesis:

Null hypothesis:

$H_0:\mu =2$

That is, the mean fuel tank capacity of all dirt bikes is not less than 2 gallons.

Alternative hypothesis:

$H_a:\mu <2$

That is, the mean fuel tank capacity of all dirt bikes is less than 2 gallons.

Determine the level of relevance.

The significance level is, in this case,$\alpha =0.10.$

Step 2: Definition

Compute the value of the test statistic by using Excel add-in (PHStat).

Excel add-in (PHStat) procedure:

Step 1: In EXCEL, Select Add-Ins > PHStat > One-Sample Tests > t Test for the mean, Sigma unknown.

Step 2: Enter 2 under Null hypothesis.

Step 3: Enter 0.10 under Level of significance.

Step 4: In Sample Statistic Options, choose Sample Statistic Known and enter 30 as the Sample size and 1.91 as the Sample mean, and 0.74 as the Sample standard deviation.

Step 5: Select Lower Tail Test from the Test Options menu.

Step 6: In Output Options, enter a Title and click OK.

Step 3: Expression

Excel add-in (PHStat) output:

t Test for Hypothesis of the Mean

Data
Null Hypothesis $\mu =$	2
Level of significance	0.1
Sample Size	30
Sample Mean	1.91
Sample Standard Deviation	0.74

Intermediate Calculations
Standard Error of the Mean	0.1351
Degrees of Freedom	29
t Test Statistic	-0.6661

Lower-Tail Test
Lower Critical Value	-1.3114
p-value	0.2553
Do not reject the null hypothesis

From the output, the value of test statistic is -0.6661 the critical value is -1.3114 and the P-value is 0.2553.

Step 4: Calculation

Critical value: From the Excel add-in (PHStat) output, the critical value is -1.3114.

P-value: From the Excel add-in (PHStat)output, the P-value is 0.2553 .

Critical value approach:

Rejection rule:

If the value of test statistic falls in the rejection region, reject the null hypothesis $H_0$.

If the values of test statistic do not falls in the rejection region, then do not reject the null hypothesis $H_0$.

Here, the value of test statistic does not falls in the rejection region. That is, $t(=-0.6661)>t_{\text{crit }}(-1.3114)$. Therefore, the null hypothesis is not rejected at $10\%$ level.

As a result, at a 10% level of significance, the test findings are not statistically significant.

Step 5: Calculation

P - value approach:

If $P\le \alpha$, then reject the null hypothesis.

The P-value is 0.2553, which is higher than the significance level. That is, $P(=0.2553)>\alpha (=0.10).$

Therefore, the null hypothesis is not rejected at $10\%$ level.

Thus, it can be conclude that the test results are not statistically significant at $10\%$ level of significance.

Interpretation:

The evidence does not support the conclusion that the average gasoline tank capacity of all dirt motorcycles is less than 2 litres.