Question

Consider the data:

54.76943, 49.78428, 59.73031, 53.00500, 52.21409, 55.76271, 51.52056, 59.62548, 52.64808, 54.54057, 49.36785, 49.87563, 59.95694, 46.19296, 43.61810, 57.60048, 55.21919, 50.46022, 47.61718, 55.50787, 61.46880, 70.16956, 48.49963, 55.30843, 67.78601, 56.49535, 46.27280, 45.97918, 52.61301, 48.00495, 58.11208, 59.43714, 51.06735, 61.98046, 47.67910, 48.73415, 54.99035, 53.13985, 47.36505, 59.45419, 49.89918, 53.75335, 42.68189, 65.78460, 50.07936, 42.35713, 64.94564, 43.36356, 54.26409, 63.13011, 56.32636, 56.04003, 49.25544, 50.66720, 56.83328, 57.16458, 59.30200, 53.86724, 53.59964, 56.99553, 50.20579, 54.16316, 54.43363, 62.48171, 58.16778, 52.14164, 53.91043, 45.11646, 60.06521, 65.60843, 53.01025, 50.47789, 59.13059, 40.51899, 59.05509, 53.77679, 43.64633, 68.95381, 58.90353, 56.35329, 48.58749, 56.25826, 55.59718, 52.98882, 56.61254, 47.25883, 63.48664, 55.34731, 55.92849, 57.44584, 45.76123, 55.32173, 56.56125, 48.08750, 47.40124, 59.04965, 54.09277, 57.76744, 54.87766, 51.85311

a. What is the 99% two-sided confidence interval for the mean of this data? _lo = _hi =

b. Compute the 95% one-sided lower confidence limit for the mean of this data. _lo =

c. Compute the 85% one-sided upper confidence limit for the mean of this data. _hi=

Answer 1

To calculate the confidence interval we need to calculate the mean and standard deviation of the given data set as:

The mean is calculated as:

Mean = (54.76943 + 49.78428 + 59.73031 + ...................+ 54.87766 + 51.85311)/100
= 5414.29036/100

Mean = 54.1429

and the standard deviation is:

s = 6.0224

a) The confidence interval is calculated as:

μ = ± t(s_M)

where:

= sample mean
t = t statistic determined by confidence level and the degree of freedom n - 1 = 100-1= 99.
s_M = standard error = √(s²/n)

so,

= 54.1429
t = 2.63 calculated using excel formula for t-distribution which takes the degree of frredom and level of significance as parameters, the formula used is =T.INV.2T(0.01, 99)
s_M = √(6.0224²/100) = 0.6

μ = M ± t(s_M)
μ = 54.1429 ± 2.63*0.6
μ = 54.1429 ± 1.581726

Thus the two-sided confidence interval is:

Lower bound = 52.5612,

Upper bound = 55.7246

b) The confidence interval at 95% confidence interval is computed as:

= 54.1429
t = 1.98 calculated using excel formula for t-distribution which takes the degree of frredom and level of significance as parameters, the formula used is =T.INV.2T(0.05, 99)
s_M = √(6.0224²/100) = 0.6

μ = ± t(s_M)
μ = 54.1429 ± 1.98*0.6
μ = 54.1429 ± 1.194975

The confidence interval is expressed as:

The lower limit is: 52.9479

c) The confidence interval at 85% confidence interval is calculated as:

= 54.1429
t = 1.45 calculated using excel formula for t-distribution which takes the degree of frredom and level of significance as parameters, the formula used is =T.INV.2T(0.15, 99)
s_M = √(6.0224²/100) = 0.6

μ = ± t(s_M)
μ = 54.1429 ± 1.45*0.6
μ = 54.1429 ± 0.0874

The confidence interval is expressed as:

The Upper limit is: 54.2303