You want to determine if your widgets from machine 1 are the same as machine 2. Machine 1 has a sample mean of 15 and a sample standard deviation 6 and a sample size of 18. Machine 2 has a sample mean of 12 and a sample standard deviation of 6 with a sample size of 18. With an alpha of .05 can we claim that there is a difference between the output of the two machines. Which of the following statements are true?
If you perform an independent groups t-test, you should be able to determine whether or not there is a difference between the two.
correct answer
To determine if there is a difference between the output of the two machines, we can perform a hypothesis test. In this case, we want to check if the means of the two samples are significantly different from each other.
The null hypothesis (H₀) states that there is no difference between the means of the two machines. The alternative hypothesis (H₁) states that there is a difference between the means of the two machines.
To perform the hypothesis test, we can use a two-sample t-test. This test compares the means of two independent samples and determines if they are significantly different.
In this case, we have the following information:
For machine 1:
Sample mean (x̄₁) = 15
Sample standard deviation (s₁) = 6
Sample size (n₁) = 18
For machine 2:
Sample mean (x̄₂) = 12
Sample standard deviation (s₂) = 6
Sample size (n₂) = 18
With an alpha (α) level of 0.05, we need to calculate the test statistic and compare it to the critical value to make a decision.
The test statistic for the two-sample t-test is calculated using the formula:
t = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂))
where:
x̄₁ and x̄₂ are the sample means
s₁ and s₂ are the sample standard deviations
n₁ and n₂ are the sample sizes
We can substitute the values into the formula:
t = (15 - 12) / √((6^2/18) + (6^2/18))
Calculating this, we find that t ≈ 0.866.
Next, we need to determine the critical value for a two-tailed test at a significance level of 0.05. Since the sample size is 18, we have 16 degrees of freedom (n₁ + n₂ - 2).
Looking up the critical value in a t-distribution table or using a statistical software, we find that the critical value for α/2 = 0.025 and 16 degrees of freedom is approximately ±2.120.
Comparing the test statistic (0.866) to the critical value (±2.120), we can see that the test statistic does not fall in the critical region. Therefore, we fail to reject the null hypothesis.
Based on these results, we conclude that there is not enough evidence to claim that there is a difference between the output of the two machines at a significance level of 0.05.