Question

In the general population, it is known that scores on a standard assessment are normally distributed with μ = 100 and σ = 14. A sample of size n = 49 is randomly selected from a new population was found to have a mean of M = 86. Calculate the 90%, 95%, and 99% confidence intervals about the M.

In the general population, it is known that scores on a personality assessment are normally distributed with μ = 120 and σ = 18. A sample of size n = 144 is randomly selected from a new population was found to have a mean of M = 100. Calculate the 90%, 95%, and 99% confidence intervals about the M.

Answer 1

a) (i) Given that the distribution is normal hence the confidence interval around the mean, M is calculated as:

μ = M ± Z(s_M)

where:

M = sample mean
Z = Z statistic determined by the confidence level using the Z table shown below
s_M = standard error = √(s²/n)

Let us assume σ=s here for the calculation

at 90% confidence level the Z score is 1.645 so,

M = 86
Z = 1.645
s_M = √(14²/49) = 2

μ = M ± Z(s_M)
μ = 86 ± 1.645*2
μ = 86 ± 3.29

90% Confidence Ievel [82.71, 89.29].

(ii) At 95% confidence interval the Z score is 1.96 so, the Confidence interval is calculated as:

M = 86
Z = 1.96
s_M = √(14²/49) = 2

μ = M ± Z(s_M)
μ = 86 ± 1.96*2
μ = 86 ± 3.92

95% Confidence Ievel [82.08, 89.92].

(iii) At 99% confidence interval the Z score is 2.58, thus the confidence interval is calculated as:

M = 86
Z = 2.58
s_M = √(14²/49) = 2

μ = M ± Z(s_M)
μ = 86 ± 2.58*2
μ = 86 ± 5.15

99% Confidence Ievel [80.85, 91.15].

b) Now changing the data just as:

M = 100, n = 144 , and  σ =s= 18

Now similarly as done in part a) the confidence interval at 90% confidence level is calculated as:

M = 100
Z = 1.645
s_M = √(18²/144) = 1.5

μ = M ± Z(s_M)
μ = 100 ± 1.645*1.5
μ = 100 ± 2.47

M = 100, 90% Confidence Ievel [97.53, 102.47].

At 95% Confidence level the interval is computed as:

M = 100
Z = 1.96
s_M = √(18²/144) = 1.5

μ = M ± Z(s_M)
μ = 100 ± 1.96*1.5
μ = 100 ± 2.94

M = 100, 95% Confidence Ievel [97.06, 102.94].

Again at a 99% confidence level:

M = 100
Z = 2.58
s_M = √(18²/144) = 1.5

μ = M ± Z(s_M)
μ = 100 ± 2.58*1.5
μ = 100 ± 3.86

M = 100, 99% [96.14, 103.86].

The Z table used for Z score computation at different confidence level is shown below as: