As part of a study of natural variation in blood chemistry s

As part of a study of natural variation in blood chemistry, serum potassium concentrations were measured in 84 randomly selected healthy women. The sample mean concentration was 4.36 mEq/l, and the sample standard deviation was 0.42 mEq/l.
(a) Find a 95% condence interval for the true serum potassium concentrate in healthy women. (b) Interpret your condence interval found in (a) in terms of the problem. (c) Does your interval support the claim that normal serum potassium concentrations are above 2.3? Explain. (d) If we were to build a 99% condence interval instead, would it widen or narrow? Explain (but do not calculate the interval).

Solution

A)

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 =    4.36          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    0.42          
n = sample size =    84          
              
Thus,              
Margin of Error E =    0.089816833          
Lower bound =    4.270183167          
Upper bound =    4.449816833          
              
Thus, the confidence interval is              
              
(   4.270183167   ,   4.449816833   ) [answer]

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b)

We are 95% confident that the true serum potassium concentrate mean in healthy women is between 4.2702 and 4.4498. [ANSWER]

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c)

YES, because the whole interval is above 2.3. [ANSWER]

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d)

It will widen, because it will need a greater critical z value, which will increase the margin of error.