Question

A company has a new process for manufacturing large artificial sapphires. In a trial run, 37 sapphires are produced. The mean weight for these 37 gems is 6.75 carats, and the sample standard deviation is 0.33 carat. Obtain a 95% confidence interval for the true mean weight of all sapphires produced by the new process. Interpret your results in the context of the problem. State the margin of er

Answer 1

The margin of error is of 0.11 carats.

The 95% confidence interval for the true mean weight of all sapphires produced by the new process is between 6.64 and 6.86 carats. The interpretation is that we are 95% sure that the true weight of all sapphires produced by the company using the new process is between these two values.

Explanation:

We have the standard deviation for the sample, which means that we use the 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 = 37 - 1 = 36

95% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 36 degrees of freedom(y-axis) and a confidence level of
`1 - (1 - 0.05)/(2) = 0.975`. So we have T = 2.0281

The margin of error is:

`M = T(s)/(√(n)) = 2.0281(0.33)/(√(37)) = 0.11`

In which s is the standard deviation of the sample and n is the size of the sample.

The margin of error is of 0.11 carats.

The lower end of the interval is the sample mean subtracted by M. So it is 6.75 - 0.11 = 6.64 carats

The upper end of the interval is the sample mean added to M. So it is 6.75 + 0.11 = 6.86 carats.

The 95% confidence interval for the true mean weight of all sapphires produced by the new process is between 6.64 and 6.86 carats. The interpretation is that we are 95% sure that the true weight of all sapphires produced by the company using the new process is between these two values.