 (10 points)Given a sample size of n = 484. Let the variance of the population be σ^{2} = 10.24. Let the mean of the sample be xbar = 20. Construct a 90% confidence interval for µ, the mean of the population, using this data and the central limit theorem.
Use Summary 5b, Table 1, Column 1
 What is the standard deviation (σ) of the population? = 3.2
 What is the standard deviation of the mean xbar of sample size n, i.e. what is σ_{xbar} , in terms of σ and n using the central limit theorem?
 Is this a onesided or twosided problem?
 What value of z should be used in computing k, the margin of error, where
z = k/σ_{xbar} ?
 What is k?
 Write the 90% confidence interval for µ based on xbar and k,
(xbar – k) < µ < (xbar + k)
 Using the Zscore applet “Area from a value”. Let the Mean = 20, and SD = σ_{xbar}. Choose “Between (xbar k) and (xbar + k)” using xbar = 20 and your computed value of k. Hit “Recalculate”. Does the probability approximately equal 0.90? (yes or no). Include a screen shot of your answer.
 (10 points) Candidate A claims he is getting a 60% positive approval rating. A recent poll has candidate A polling a 55% approval rating based on a sample of 600 voters (meaning 330 voters said they were approving of candidate A). Construct a 95% proportion confidence interval based on this data and determine whether the data supports the 60% claim.
Start by using the binomial distribution ( + )^{n} = 1, and = n.
Use Summary 5b, Table 1, Column 3.
 What is n? = 600
 What is ? =
 What is ? =
 What is ? =
 What is p? (Note: p is based on our belief about the population. is based on the sample.) =
 Let σ_{p} = sqrt(). What is σ_{p} ?
 We want to make a confidence interval by using the formula
z = k/σ_{p}
Should we use a 1 tail or 2 tail zscore?
 What value of z corresponds to the desired 95% confidence interval?
 What is k?
 Construct the confidence interval
( – k) < p_{true}< ( + k)
 Is your confidence intervalconsistent with the belief that 60% approval rating claim? (yes or no, and explain your answer using your computed confidence interval).
Type of sample  Population parameters  Sample statistics 
General  µσ N  s n 
Binomial (proportion)  p σ N  σ or s n 
Note: A “General” sample means that the values in the population can be any number between inf and +inf. A “Binomial” sample means that the values in the population can take on only 2 values like: p/not p, heads/tails, black/white, male/female, on/off, 1/0, filled/empty, etc.
Table 1: Confidence Intervals
σ is known – General sample  σ is unknown – General sample  σ is approximately known – Binomial sample 
σ n  s n  σ = 
z = k/[σ/]  t = k/[s/]  z = 
k = z*σ/
where z = 1.645, 1.96, 2.576 (90%,95%,99%) 
k = t*s/
find t with tdistribution applet using: 2 tails, df = (n – 1), p = .10, .05, .01 
k = z*
where z = 1.645, 1.96, 2 .576 
( – k) <µ< ( + k) 
( – k) <µ< ( + k) 
( – k) < p < ( + k)

Table 2: Statistical Comparison of Mean Values
General sample vs theoretical value  General sample vs General sample  Binomial sample vs theoretical value  Binomial sample vs Binomial sample 
Ho: µ = constant
Ha: µ> constant 
Ho: µ_{1} = µ_{2}
Ha: µ_{1}>µ_{2} 
Ho: p = constant
Ha: p > constant 
Ho: p_{1} = p_{2}
Ha: p_{1}> p_{2} 
s n  s_{1}n_{1} s_{2} n_{2}
s = 
s =  s_{1} = s_{2} =
s = 
t =  t =  t =  t = 
df = n – 1
right tail 
df = (n_{1} – 1) + (n_{2} – 1)
right tail 
df = n 1
right tail 
df = (n_{1} – 1) + (n_{2} – 1)
right tail 
Using the tdistribution applet, find the probability p. (Don’t confuse this p with the proportion constant in Table 2, Column 3!)
Compare p with α.
If p >α, accept Ho.
If p <α, reject Ho, accept Ha instead.
Last Updated on