The cycle times for an electroplating machine are believed to be normally distributed. Twenty-one plating operations are randomly sampled for time to cycle. The sample average cycle time is 10.21 minutes with a standard deviation of 0.59 minutes. Construct a 95% confidence interval on the mean cycle time for this machine.

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25-04-2020
Mathematics

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The cycle times for an electroplating machine are believed to be normally distributed. Twenty-one plating operations are randomly sampled for time to cycle. The sample average cycle time is 10.21 minutes with a standard deviation of 0.59 minutes. Construct a 95% confidence interval on the mean cycle time for this machine.

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joaobezerra joaobezerra · Answer 1 · 2020-04-27T02:23:49+02:00

Answer:

The 95% confidence interval on the mean cycle time for this machine is between 8.98 minutes and 11.44 minutes.

Step-by-step explanation:

We are in posession of the sample's standard deviation, so we use the students' t-distribution to solve this question.

The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So

df = 21 - 1 = 20

95% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 20 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.95}{2} = 0.975([tex]t_{975}[/tex]). So we have T = 2.086

The margin of error is:

M = T*s = 2.086*0.59 = 2.13

In which s is the standard deviation of the sample.

The lower end of the interval is the sample mean subtracted by M. So it is 10.21 - 1.23 = 8.98 minutes

The upper end of the interval is the sample mean added to M. So it is 10.21 + 1.23 = 11.44 minutes

The 95% confidence interval on the mean cycle time for this machine is between 8.98 minutes and 11.44 minutes.