Z* confidence intervals

Question

If n = 420 and  = 0.45, construct a 95% confidence interval.

Answer 1

The calculator solution would be using 1PropZInt ->

x: .45*420 = 189

n: 420

C-Level: .95

Then Calculate -> Giving you the CI of {.40242, .49758}

Answer 2

First we use our confidence interval of 95% and transfer that to a decimal to get 0.95

This represents 95% of the normal bell curve, meaning there are two tails left and no covered by that 95%

see picture at the website: https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/confidence-interval/

for visual

To find the remaining part of the curve we need to take 100%-confidence interval percent or 1-decimal confidence interval

1-0.95=0.05 area under the curve left

This is the total area left so to get it in terms of tales: upper and lower limit or right and left tale we need to divide by two

0.05/2 = 0.025 left sided area and 0.025 right sided area

Now we take our normal standard probabilities or z-table

At the top of the table is decimals between 0.00 to 0.09

The leftmost column has numbers from -3.4 to 3.4

Let us take the left area under the curve or our lower limit--- this would be a negative value in z as it is less than our xbar or mean

So we start on the leftmost column and go down in the negative numbers and we scroll across to the middle of the table until we see our alpha value or 0.025. Once we find our alpha value we then scroll up until we get our decimal on the top row.

For our example we have -1.9 and 7 columns over we find 0.0250. At the top of this row we find 0.06 so our z-value for 0.0250 (left or lower limit) is -1.96 (-1.9 + -0.06)

If we follow this sample method we can get the upper limit or right sided tail. Again we look for the 0.0250 in the middle of the table but we look at the positive numbers this time

We get 1.96

Now that you have your z-values you can find the sample statistic using this formula:

Sample statistic + / - z*SE

Where SE is the standard error , z is your z-score (critical z-score or the absolute value of your z-score) score) and your sample statistic is a range of values

Sample Statistic +/- 1.96 *SE

SE =sqrt[ (phat *(1-phat)) / n where n is our sample size

SE=sqrt[ (0.45*(1-0.45))/ 420] = 0.024275

Sample Statistic = ( 0.45 + 1.96*0.24275 , 0.45-1.96*0.24275)

Sample Statistic = ( 0.92579 , -0.02579)