a. A statistics practitioner randomly sampled 100 observations from a population with a standard deviation of 5 and found that x is 10. Estimate the population mean with 90% confidence.
b. Repeat part (a) with a sample size of 25.
c. Repeat part (a) with a sample size of 10.
d. Describe what happens to the confidence interval estimate when the sample size decreases.
a. n = 100
σ = 5
"\\bar{x}=10"
For the confidence level 1 – α = 0.90. determine "t_{\u03b1\/2} = t_{0.05}" using t table, which is given in the row with df = n-1 = 100-1 = 99 and in the column with "t_{0.05}" :
"t_{\u03b1\/2} = 1.64"
The margin of error is then:
"E = t_{\u03b1\/2} \\times \\frac{\u03c3}{\\sqrt{n}} \\\\\n\nE = 1.64 \\times \\frac{5}{\\sqrt{100}} = 0.82"
The boundaries of the confident interval:
10 ± 0.82 (9.18 to 10.82)
b. n = 25
For the confidence level 1 – α = 0.90. determine "t_{\u03b1\/2} = t_{0.05}" using t table, which is given in the row with df = n-1 = 25-1 = 24 and in the column with "t_{0.05}" :
"t_{\u03b1\/2} = 1.71"
The margin of error is then:
"E = t_{\u03b1\/2} \\times \\frac{\u03c3}{\\sqrt{n}} \\\\\n\nE = 1.71 \\times \\frac{5}{\\sqrt{25}} = 1.71"
The boundaries of the confident interval:
10 ± 1.71 (8.29 to 11.71)
c. n=10
For the confidence level 1 – α = 0.90. determine "t_{\u03b1\/2} = t_{0.05}" using t table, which is given in the row with df = n-1 = 10-1 = 9 and in the column with "t_{0.05}" :
"t_{\u03b1\/2} = 1.83"
The margin of error is then:
"E = t_{\u03b1\/2} \\times \\frac{\u03c3}{\\sqrt{n}} \\\\\n\nE = 1.83 \\times \\frac{5}{\\sqrt{10}} = 2.89"
The boundaries of the confident interval:
10 ± 2.89 (7.11 to 12.89)
d. Notice how the interval increases and the sample size decreases
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