using 100 jellie beans construct a hypothetical 95 confidenc

using 100 jellie beans construct a hypothetical 95% confidence interval for a hypothethical case. Use your own unique choice of mean, standard deviation, and sample size calculate the confidence interval based on the confidence changes to 90%, the confidence changes to 99%, the sample size is cut in half, the sample size is doubled and the sample size is tripled.

Solution

Let mean weight of each jelly bean (X) = 1.5 grams

sample standard deviation of weights = 0.4 grams

Here, the initial sample size is n = 100.

a)

95% confidence:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    1.5          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    0.4          
n = sample size =    100          
              
Thus,              
Margin of Error E =    0.078398559          
Lower bound =    1.421601441          
Upper bound =    1.578398559          
              
Thus, the confidence interval is              
              
(   1.421601441   ,   1.578398559   ) [ANSWER]

*********************

b)

90% confidence:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.05          
X = sample mean =    1.5          
z(alpha/2) = critical z for the confidence interval =    1.644853627          
s = sample standard deviation =    0.4          
n = sample size =    100          
              
Thus,              
Margin of Error E =    0.065794145          
Lower bound =    1.434205855          
Upper bound =    1.565794145          
              
Thus, the confidence interval is              
              
(   1.434205855   ,   1.565794145   ) [ANSWER]

***********************

c)

99% confidence:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.005          
X = sample mean =    1.5          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    0.4          
n = sample size =    100          
              
Thus,              
Margin of Error E =    0.103033172          
Lower bound =    1.396966828          
Upper bound =    1.603033172          
              
Thus, the confidence interval is              
              
(   1.396966828   ,   1.603033172   ) [ANSWER]

***********************

d)

Sample size cut in half, so n = 50:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    1.5          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    0.4          
n = sample size =    50          
              
Thus,              
Margin of Error E =    0.110872306          
Lower bound =    1.389127694          
Upper bound =    1.610872306          
              
Thus, the confidence interval is              
              
(   1.389127694   ,   1.610872306   ) [ANSWER]

*************************

e)

Sample size is doubled, so n = 200:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    1.5          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    0.4          
n = sample size =    200          
              
Thus,              
Margin of Error E =    0.055436153          
Lower bound =    1.444563847          
Upper bound =    1.555436153          
              
Thus, the confidence interval is              
              
(   1.444563847   ,   1.555436153   ) [ANSWER]

***************************

f)

Sample size is tripled (n = 300):

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    1.5          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    0.4          
n = sample size =    300          
              
Thus,              
Margin of Error E =    0.045263429          
Lower bound =    1.454736571          
Upper bound =    1.545263429          
              
Thus, the confidence interval is              
              
(   1.454736571   ,   1.545263429   ) [ANSWER]