An engineering company believes it has developed a faster way to complete the assembly of an industrial machine. The present process takes an average of 6.5 hours to complete and the times it takes to complete the process are approximately normally distributed. If the mean time for the new process is faster at the 0.05 level of significance, the engineering company will present the new process to their clients. Otherwise, the company will do further research. The company assembles a number of machines using the new process and randomly selects 17. The assembly times for the 17 machines in the sample are shown. Perform a hypothesis test showing your work.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To perform a hypothesis test, we need to state the null hypothesis (H0) and the alternative hypothesis (Ha). In this case, the null hypothesis is that the mean assembly time of the new process is equal to the mean assembly time of the current process. The alternative hypothesis is that the mean assembly time of the new process is faster than the mean assembly time of the current process.
Let's denote µ1 as the mean assembly time of the current process and µ2 as the mean assembly time of the new process.
H0: µ1 - µ2 = 0 (No difference in mean assembly time)
Ha: µ1 - µ2 > 0 (Mean assembly time of the new process is faster)
We will use a one-tailed hypothesis test with a significance level (α) of 0.05.
Next, we need to calculate the test statistic and determine the critical value. Since the sample size is relatively small (n=17), we will use a t-test instead of a z-test.
1. Calculate the sample mean and standard deviation:
Let's denote x̄ as the sample mean and s as the sample standard deviation.
x̄ = (sum of assembly times)/(number of samples)
s = sqrt((sum of (assembly time - x̄)^2)/(number of samples - 1))
2. Calculate the t-statistic:
t = (x̄ - µ1) / (s / sqrt(n))
The test statistic follows a t-distribution with (n-1) degrees of freedom.
3. Determine the critical value:
To determine the critical value, we need to find the t-value from the t-distribution table (or use statistical software).
The critical value is the value beyond which we reject the null hypothesis.
The critical value for a one-tailed test with α=0.05 and (n-1) degrees of freedom can be found in the t-distribution table.
4. Compare the test statistic with the critical value:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
By performing these calculations and comparisons, we can determine whether the new process is faster than the current process, and whether the engineering company should present it to their clients.