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# Standard D and Confidence Intervals

1. (10 points)Given a sample size of n = 484. Let the variance of the population be σ2 = 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

1. What is the standard deviation (σ) of the population? = 3.2
2. What is the standard deviation of the mean xbar of sample size n, i.e. what is σxbar , in terms of σ and n using the central limit theorem?
3. Is this a one-sided or two-sided problem?
4. What value of z should be used in computing k, the margin of error, where

z = k/σxbar ?

1. What is k?
2. Write the 90% confidence interval for µ based on xbar and k,

(xbar – k) < µ < (xbar + k)

1. Using the Z-score applet “Area from a value”. Let the Mean = 20, and SD = σxbar. 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.

1. (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.

1. What is n? = 600
2. What is ? =
3. What is ? =
4. What is ? =
5. What is p? (Note: p is based on our belief about the population. is based on the sample.) =
6. Let σp = sqrt(). What is σp ?
7. We want to make a confidence interval by using the formula

z = k/σp

Should we use a 1 tail or 2 tail z-score?

1. What value of z corresponds to the desired 95% confidence interval?
2. What is k?
3. Construct the confidence interval

( – k) < ptrue< ( + k)

1. 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 t-distribution 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 = constantHa: µ> constant Ho: µ1 = µ2Ha: µ1>µ2 Ho: p = constantHa: p > constant Ho: p1 = p2Ha: p1> p2 s  n s1n1  s2  n2 s = s = s1 = s2 =s = t = t = t = t = df = n – 1right tail df = (n1 – 1) + (n2 – 1)right tail df = n -1right tail df = (n1 – 1) + (n2 – 1)right tail

Using the t-distribution 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 November 12, 2019

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