Question

In 36 randomly selected seawater samples, the mean sodium chloride concentration was 23 cubic centimeters per cubic meter. Assume sample standard deviation is 6.7 cubic centimeters per cubic meter. Construct a 95% and 90% confidence interval for the population mean.

Answer 1

Answer:

The 90% confidence interval for the population mean is between 11.68 and 34.32 cubic centimeters per cubic meter.

The 95% confidence interval for the population mean is between 9.4 and 36.6 cubic centimeters per cubic meter.

Step-by-step explanation:

We are in posession of the sample 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 = 36 - 1 = 35

90% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 35 degrees of freedom(y-axis) and a confidence level of $1 - \frac{1 - 0.9}{2} = 0.95([tex]t_{95}$). So we have T = 1.6896.

The margin of error is:

M = T*s = 1.6896*6.7 = 11.32.

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 23 - 11.32 = 11.68 cubic centimeters per cubic meter.

The upper end of the interval is the sample mean added to M. So it is 23 + 11.32 = 34.32 cubic centimeters per cubic meter.

The 90% confidence interval for the population means is between 11.68 and 34.32 cubic centimeters per cubic meter.

95% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 35 degrees of freedom(y-axis) and a confidence level of $1 - \frac{1 - 0.95}{2} = 0.975([tex]t_{97.5}$). So we have T = 2.0301.

The margin of error is:

M = T*s = 2.0301*6.7 = 13.60.

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 23 - 13.60 = 9.4 cubic centimeters per cubic meter.

The upper end of the interval is the sample mean added to M. So it is 23 + 13.60 = 36.6 cubic centimeters per cubic meter.

The 95% confidence interval for the population means is between 9.4 and 36.6 cubic centimeters per cubic meter.