1) Write out the formula for calculating the 95% confidence interval for a sampl

Question

1) Write out the formula for calculating the 95% confidence interval for a sample mean.

2) In this problem, you will compute confidence intervals for the mean under a variety of conditions and compare the results. You will also discuss what accounts for differences in these confidence intervals, if any. Please use Excel to show your work and report your results with two digits to the right of the decimal point.

a) Using =T.INV.2T() and the formula for calculating the confidence interval, calculate the exact 95% confidence interval (upper and lower limits) for the following:

i) n = 40, mean = 32, S.D. = 6

ii) n = 40, mean = 32, S.D. = 12

iii) n = 140, mean = 32, S.D. = 6

iv) n = 140, mean = 32, S.D. = 12

b) Using t = 2 and the formula for calculating the confidence interval, calculate the approximate 95% confidence interval (upper and lower limits) for the following:

i) n = 40, mean = 32, S.D. = 6

ii) n = 40, mean = 32, S.D. = 12

c) Compare the confidence intervals computed in a.i and a.ii, above. What accounts for any differences observed?

d) Compare the confidence intervals computed in a.i and a.iii, above. What accounts for any differences observed?

e) Compare the confidence intervals computed in b.i and b.ii, above. What accounts for any differences observed?

f) Compare the confidence intervals computed in a.i and b.i, above. What accounts for any differences observed?

Explanation / Answer

1)

CI = mean+ / -z * (std.deviation/ sqrt(n))

= mean + - 1.96 * (std.deviation /sqrt(n))

Here, z value for 95% = 1.96

a)

b)

c)
As std.deviation increases confidence interval increases

d)
As sample size increases confidence interval increases

e)

As std.deviation increases confidence interval increases

f)
There is no difference approx slight difference

CI for 95% 95% 95% 95% n 40 40 140 140 mean 32 32 32 32 t-value of 95% CI 2.0227 2.0227 1.9772 1.9772 std. dev. 6 12 12 12 SE = std.dev./sqrt(n) 0.94868 1.89737 1.01419 1.01419 ME = t*SE 1.91889 3.83779 2.00522 2.00522 Lower Limit = Mean - ME 30.08111 28.16221 29.99478 29.99478 Upper Limit = Mean + ME 33.91889 35.83779 34.00522 34.00522 95% CI (30.0811 , 33.9189 ) (28.1622 , 35.8378 ) (29.9948 , 34.0052 ) (29.9948 , 34.0052 )

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