Help with confidence intervals, please.?

Question

In an article in the Journal of Management, Morris, Avila and Allen studied innovation by surveying firms to find the number of new products introduced by the firms. A randon sample of 100 California firms are selected. Each firm is asked to report the number of new products introduced last year. The survey found that on average, these firms introduced 5.68 products with a standard deviation of 8.70. Compute a 98% confidence interval for the new products introduced last year.

Answer 1

The formula for a confidence interval:

CI = xbar ± z * (s / sqrt(n))
where:
CI = Confidence Interval
xbar = sample mean (5.68)
s = sample standard deviation (8.70)
n = number of observations (100)
z = 2.325 (from table lookup)

Since n > 30, we assume a normal distribution and the sample standard deviation can be substituted for the population standard deviation. (This means using a z-test instead of a t-test).

The confidence interval is :
CI = xbar ± z * (s / sqrt(n))
CI = 5.68 ± 2.325 * (8.70 / sqrt(100))
CI = 5.68 ± 2.325 * (8.70 /10)
CI = 5.68 ± 2.325 * (.87)
CI = 5.68 ± 2.02
CI = 3.66 to 7.70

Good luck in your studies,
~ Mitch ~

Answer 2

This problem sounds unusual. Make sure you haven't misplaced a decimal point.
If you have a TI-83 Plus or TI-84 calculator, you can do it this way:
a) Press STAT, cursor to TESTS, and press 7.
b) On the screen that appears, cursor to "Stats" on the ZInterval screen and press ENTER.
c) Enter data opposite positions as follows: σ: 8.7, x¯ :5.68, n:100, and C-Level: .98.
d) Cursor down to Calculate, press ENTER, and the interval (3.6561, 7.7039) will appear along with the values for "n" and the mean.

If you don’t have a calculator you need to solve this:
u = x-bar +- E where u is, mu, the population mean, x-bar is the sample mean and E is the error.
u = x-bar +- Z(a/2) sigma/sqrt (n)
First we need to ook up Z(a/2) where a/2 is .02/2 = .01
Now, the are to ghr right of the z-value is .5-0.1. = 0.49
Consulting a set of z-values, we find Z = 2.33

Now going back to subjstitute numbers in the formmula:
E = Z(a/2) *sigma/sqrt(n)
=2.33*8.7/sqrt(100)
= 2.33*.87
= 2.027
So, we have
u = x-bar +- Z(a/2) sigma/sqrt (n)
u = 5.68 +- 2.027
I will leave that arithmetic to you.

Hope this helps.
FE

Answer 3

Alpha value for 98% is 1.0-.98=.02 - You have to look up a test statistic for your given information in a table or using an online calculator.

T-test for the given information = 2.364
The equation for the CI is..
mean +/- Ttest(standard_dev/ sqrt(sample_size)
5.68 +/- 2.364(8.7/sqrt(100)) = 5.68 +/- 2.057

The 98% CI is then 5.68 +/- 2.057

Shortcut - type the following into an Excel spreadsheet cell.
=confidence(.02,8.7,100)

That will give you the +/- value to add and subtract from the mean to get the interval.

Answer 4

alpha=0.95 dividing 95 by 100.given fromconfidence interval confidence level = (1 - 0.95) = 0.05 = 5%. A.