You intend to estimate a population mean

with the following sample.
77
44.4
33.6
63.1
68.1
70.6
52
43.5
26.4
72.6
30.9
29.4

You believe the population is normally distributed. Find the 90% 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).

90% C.I. =

Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

1 answer

To find the 90% confidence interval for the population mean based on the given sample, we will follow these steps:

  1. Calculate the sample mean (x¯).
  2. Calculate the sample standard deviation (s).
  3. Determine the critical value (t) for a 90% confidence level.
  4. Calculate the margin of error.
  5. Construct the confidence interval.

Step 1: Calculate the sample mean (x¯)

The sample data is: 77,44.4,33.6,63.1,68.1,70.6,52,43.5,26.4,72.6,30.9,29.4

First, calculate the sum of the sample values:

Sum=77+44.4+33.6+63.1+68.1+70.6+52+43.5+26.4+72.6+30.9+29.4=683.8

Then, the sample mean (x¯) is:

x¯=Sumn=683.81256.9833

Step 2: Calculate the sample standard deviation (s)

We will first calculate the sum of squared deviations from the mean:

Sum of Squared Deviations=(xix¯)2

Each term (xix¯)2 is calculated as follows:

  • For 77: (7756.9833)2400.6338
  • For 44.4: (44.456.9833)2158.4526
  • For 33.6: (33.656.9833)2568.6851
  • For 63.1: (63.156.9833)237.8409
  • For 68.1: (68.156.9833)2125.8235
  • For 70.6: (70.656.9833)2178.1312
  • For 52: (5256.9833)24.8311
  • For 43.5: (43.556.9833)2179.8831
  • For 26.4: (26.456.9833)2926.0673
  • For 72.6: (72.656.9833)2236.0750
  • For 30.9: (30.956.9833)2679.5150
  • For 29.4: (29.456.9833)2743.0652

Now summing these squared deviations:

Total=400.6338+158.4526+568.6851+37.8409+125.8235+178.1312+4.8311+179.8831+926.0673+236.0750+679.5150+743.06523140.0968

Now, calculate the sample variance (s2):

s2=Sum of Squared Deviationsn1=3140.09681213140.096811285.463

Then take the square root to find the sample standard deviation (s):

s285.46316.91

Step 3: Determine the critical value (t)

For a 90% confidence level with n1=11 degrees of freedom, we look up t in the t-distribution table.

t0.05,111.796 (rounded to 3 decimal places)

Step 4: Calculate the margin of error (ME)

The margin of error is calculated as:

ME=t×sn=1.796×16.91121.796×4.886=8.796

Step 5: Construct the confidence interval

Finally, the confidence interval is given by:

(x¯ME,x¯+ME)=(56.98338.796,56.9833+8.796)

Calculating the endpoints:

Lower Limit=56.98338.79648.1873 Upper Limit=56.9833+8.79665.7793

Thus, the 90% confidence interval is:

90% C.I.=(48.19,65.78)

Final result with two decimal places as requested:

90% C.I.=(48.19,65.78)