A manufacturer of toasters claims that the toasters have a mean life of 9 years and a standard deviation of 2.1 years. A random sample of 60 such toasters is selected for testing. If the sample produces a mean value of 8.3 years, is there sufficient evidence that the mean toaster life is less than the manufacturer claimed?
Assume all samples are simple random samples and α (alpha) is taken to be 0.05.
To determine if there is sufficient evidence to support the claim that the mean toaster life is less than the manufacturer claimed, a one-sample t-test can be conducted.
The null hypothesis (H0) is that the mean toaster life is 9 years.
The alternative hypothesis (Ha) is that the mean toaster life is less than 9 years.
Given:
Sample mean (x̄) = 8.3 years
Sample size (n) = 60
Population standard deviation (σ) = 2.1 years
The test statistic for a one-sample t-test is calculated as:
t = (x̄ - μ) / (σ / √n)
Plugging in the values:
t = (8.3 - 9) / (2.1 / √60)
t = -0.7 / (2.1 / √60)
Calculating the standard error (SE):
SE = σ / √n
SE = 2.1 / √60
Calculating the degrees of freedom (df):
df = n - 1
df = 60 - 1 = 59
Now, let's calculate the t-value:
t = -0.7 / SE
Using a t-table or a calculator, we find that the critical t-value for a one-tailed test at α = 0.05 and df = 59 is approximately -1.671.
Since the calculated t-value (-0.7 / SE) of the sample is greater than the critical t-value (-1.671), we fail to reject the null hypothesis.
Therefore, there is not sufficient evidence to support the claim that the mean toaster life is less than the manufacturer claimed.