You want to determine if your widgets from machine 1 are the same as machine 2. Machine 1 has a sample mean of 5 and a population standard deviation 1 and a sample size of 4. Machine 2 has a sample mean of 8 and a population standard deviation of 2 with a sample size of 4. 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?
none
7 A company has three machines A, B and C which all produce the same two parts, X and
Y. of all the parts produced, machine A produces 60%, machine B produces 30%, and
machine C produces the rest. 40% of the parts made by machine A are part X, 50% of the
parts made by machine B are part X, and 70% of the parts made by machine C are part X.
A part produced by this company is randomly sampled and is determined to be an X part.
With the knowledge that it is an X part, find the probabilities that the part came from
machine A, B or C.
To determine if there is a difference between the output of the two machines, we can perform a hypothesis test.
The null hypothesis (H0) states that there is no difference between the output of the two machines, while the alternative hypothesis (H1) states that there is a difference. In this case, we can use a two-sample t-test to compare the means of the two samples.
Given:
Machine 1: Sample mean (x̄1) = 5, Population standard deviation (σ1) = 1, Sample size (n1) = 4
Machine 2: Sample mean (x̄2) = 8, Population standard deviation (σ2) = 2, Sample size (n2) = 4
We can calculate the test statistic using the formula:
t = (x̄1 - x̄2) / √[(s1^2/n1) + (s2^2/n2)]
where s1 and s2 are the sample standard deviations calculated from the samples.
From the given information, we can't directly calculate the sample standard deviations because we only have the population standard deviations. However, we can estimate them using the following formula:
s1 = σ1 / √n1
s2 = σ2 / √n2
Substituting the values, we get:
s1 = 1 / √4 = 1/2 = 0.5
s2 = 2 / √4 = 2/2 = 1
Now, calculating the t-test statistic:
t = (5 - 8) / √[(0.5^2/4) + (1^2/4)]
= -3 / √[0.0625 + 0.25]
= -3 / √0.3125
= -3 / 0.559
Next, we need to compare this calculated t-value with the critical t-value to determine if we reject or fail to reject the null hypothesis.
The critical t-value depends on the sample size and the chosen significance level (alpha). In this case, alpha is 0.05 (or 5%).
Since the sample size is small (n1 = n2 = 4) and we want to compare the means of two unrelated samples, we will use a two-tailed t-test. The critical t-values for alpha = 0.05 and degrees of freedom (df) = 6 (n1 + n2 - 2) are approximately ±2.447.
If the calculated t-value falls outside of this range (t < -2.447 or t > 2.447), we can reject the null hypothesis in favor of the alternative hypothesis. If the calculated t-value falls within this range, we fail to reject the null hypothesis.
Comparing the calculated t-value (-3 / 0.559 ≈ -5.36) with the critical t-values, we can see that it falls outside the range. Therefore, we can claim that there is a difference between the output of the two machines.
The true statements based on the analysis are:
1. We can claim that there is a difference between the output of the two machines.
2. The calculated t-value falls outside the range of the critical t-values for alpha = 0.05 and df = 6.
Please note that this explanation assumes that the two samples are independent and represent random samples from their respective populations.