We wish to verify at 20% significance the hypothesis that the true average gasoline consumption in a certain type of engine is 14 gallons per minute, with the alternative that it is not 14 gallons. If the tests carried out on a random sample of 10 engines have shown average consumption of 17 gallons per minute with a standard deviation of 4 gallons, can the hypothesis be ruled out? Because?

To determine if the hypothesis that the true average gasoline consumption is 14 gallons per minute can be ruled out, we will perform a one-sample t-test.

Given:
- Null hypothesis, H_0 : True average gasoline consumption is 14 gallons per minute.
- Alternative hypothesis, H_1 : True average gasoline consumption is not 14 gallons per minute.
- Sample mean, \bar{x} = 17 gallons per minute
- Sample size, n = 10
- Population standard deviation, \sigma = 4
- Level of significance, \alpha = 0.20

Calculating the t-score:
t = \frac{\bar{x} - \mu}{s/\sqrt{n}}
where \mu = 14 , s = 4 , and n = 10 .
So,
t = \frac{17 - 14}{4/\sqrt{10}} \approx 3.354

Degrees of freedom, df = n - 1 = 10 - 1 = 9

Now, we compare the calculated t-value to the critical t-value at the 20% significance level with 9 degrees of freedom from the t-distribution table. Since it is a two-tailed test, we will find t_{crit} :
t_{crit}\approx\pm1.833

Since |3.354| > 1.833 We reject the null hypothesis.

Answer:
The hypothesis that the true average gasoline consumption in a certain type of engine is 14 gallons per minute can be ruled out at a 20% significance level because the sample data provides sufficient evidence to reject the null hypothesis.