An inventor has developed a new, energy-efficient lawn mower
engine. He claims that the engine will run continuously for more
than 5 hours (300 minutes) on a single gallon of regular
gasoline. (The leading brand lawnmower engine runs for 300
minutes on 1 gallon of gasoline.)
From his stock of engines, the inventor selects a simple random
sample of 50 engines for testing. The engines run for an average
of 305 minutes. The true standard deviation σ is known and is
equal to 30 minutes, and the run times of the engines are
normally distributed.
Test hypothesis that the mean run time is more than 300
minutes. Use a 0.05 level of significance.
To test the hypothesis that the mean run time of the inventor's engines is more than 300 minutes, we will use a one-sample t-test.
Step 1: State the null and alternative hypothesis:
Null hypothesis (H0): The mean run time is equal to or less than 300 minutes.
Alternative hypothesis (Ha): The mean run time is more than 300 minutes.
Step 2: Determine the level of significance (α):
The given significance level is 0.05.
Step 3: Calculate the test statistic:
Since we know the population standard deviation (σ), we can use the one-sample t-test formula:
t = (x̄ - μ) / (σ / √n)
Where:
x̄ = sample mean
μ = population mean
σ = population standard deviation
n = sample size
In this case, x̄ = 305, μ = 300, σ = 30, and n = 50.
t = (305 - 300) / (30 / √50)
= 5 / (30 / 7.071)
= 5 / 1.414
≈ 3.54
Step 4: Calculate the critical value:
Since this is a one-tailed test (we are testing if the mean is more than 300), we will compare the test statistic to the critical value from the t-distribution.
df = n - 1 = 50 - 1 = 49
Using a t-table or calculator, we find that the critical value at α = 0.05 and df = 49 is approximately 1.677.
Step 5: Make a decision:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the test statistic (3.54) is greater than the critical value (1.677), so we reject the null hypothesis.
Step 6: State the conclusion:
Based on the sample data and the results of the one-sample t-test, there is enough evidence to support the claim that the mean run time of the inventor's engines is more than 300 minutes.