You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:

Question

You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:

39.8
41.4
39.6
31.4
32
34
21.4
33.9
39.6
42
45.8
38.6

Find the 95% confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place).

95% C.I. = 

Answer 1

In this problem we are estimating the mean from a sample of n = 12. Because the sample size is small, we have to use a t distribution for our Confidence Interval.

The formula for a (1-α)CI is as follows:

Confidence Interval = x_bar +- t_n-1(α) * (s/(√n))

1. x_bar is the sample mean.
2. s is the sample standard deviation.
3. n is the sample size
4. the "n-1" part next to the t represents the degrees of freedom necessary for looking up a value in the t-distribution table.
5. α is related to the % of the confidence interval. We have (1-α)% CI, so for a 95% CI our α is 5% or .05 (.05 for two-tailed test in the table).

The first step is to calculate the sample mean.

1. Add all of the data values and divide by n.
2. (∑ x_i)/n
3. You should obtain a value of x_bar = 36.625.

The next step is to find the sample standard deviation s. We will do this by first finding the sample variance s².

1. s² = (∑(x_i-x_bar)²) / (n-1)
2. To do this by hand, take the first value from the data and subtract the sample mean (x_bar). Square this value. Example: (39.8 - 36.625)² would be the first one, (41.4-36.625)² would be the second one, and so on.
3. Repeat this process and add up all of the values that you've obtained. Divide this sum by n-1 (in this case n-1 = 11)
4. You should obtain a final value of s² = 42.0395
5. Take the square root to find s. s = 6.483283

Next we need our t- value.

1. Our degrees of freedom is n-1 = 11.
2. Our α here is .05
3. Look in a t-table. Go down to the appropriate degrees of freedom row, and go over to the appropriate α column. In the t-table this would be df=11, α =.05 (two tailed), which gives us a value of t = 2.201

Now that we have everything we just plug it into the formula to generate the Confidence Interval

Confidence Interval = x_bar +- t_n-1(α) * (s/(√n))

= 36.625 +- 2.201* (6.483283/(√12)

= 36.625 +- 4.119309266

Subtract 4.119309266 from the mean to obtain the lower bound. Add 4.119309266 to the mean to obtain the upper bound.

95% CI = (32.51, 40.74)