Standard D and Confidence Intervals

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  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 = constant

Ha: µ> constant

Ho: µ1 = µ2

Ha: µ12

Ho: p = constant

Ha: p > constant

Ho: p1 = p2

Ha: p1> p2

 s  n   s1n1  s2  n2

 

s =

s = s1 = s2 =

s =

t = t = t = t =
df = n – 1

right tail

df = (n1 – 1) + (n2 – 1)

right tail

df = n -1

right 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.

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