Consider a normal population with an unknown population standard deviation. A ra

Question

Consider a normal population with an unknown population standard deviation. A random sample results in 1formula80.mml = 43.92 and s2 = 17.64. Use Table 2.

a. Compute a 95% confidence interval for if 1formula80.mml and s2 were obtained from a sample of 26 observations. (Round intermediate calculations to 4 decimal places, "t" value to 3 decimal places, and final answers to 2 decimal places.)

Confidence interval ___ to _____

Use your answers to discuss the impact of the sample size on the width of the interval.

Compute a 95% confidence interval for if and s2 were obtained from a sample of 29 observations. (Round intermediate calculations to 4 decimal places, "t" value to 3 decimal places, and final answers to 2 decimal places.)

Explanation / Answer

a)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    43.92          
t(alpha/2) = critical t for the confidence interval =    2.06          
s = sample standard deviation =    4.2          
n = sample size =    26          
df = n - 1 =    25          
Thus,              
Margin of Error E =    1.696796801          
Lower bound =    42.2232032          
Upper bound =    45.6167968          
              
Thus, the confidence interval is              
              
(   42.22   ,   45.62   ) [ANSWER]

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b)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    43.92          
t(alpha/2) = critical t for the confidence interval =    2.048          
s = sample standard deviation =    4.2          
n = sample size =    29          
df = n - 1 =    28          
Thus,              
Margin of Error E =    1.597277021          
Lower bound =    42.32272298          
Upper bound =    45.51727702          
              
Thus, the confidence interval is              
              
(   42.32   ,   45.52   ) [ANSWER]

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c)

As we can see, greater sample size yields narrower confidence interval, so

OPTION B: The bigger sample size will lead to a smaller interval width and therefore a more precise interval. [ANSWER]

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