Question

A random sample of 83 investment portfolios managed by Kendra showed that 62 of them met the targeted annual percent growth. Find a 99% confidence interval for the proportion of the portfolios meeting target goals managed by Kendra. a. Write the formula you will be using in the space below (1 point): b. Identify all corresponding values associated with your symbols in the space below (2 points) c. Plug in your values and show your work in the space below (3 points) d. Sketch your distribution, identify your endpoints and label them, shade the area in which your population mean would fall (1.5 points) e. Interpret your results within the context of the problem (1.5 points)

Answer 1

A random sample of 83 investment portfolios managed by Kendra showed that 62 of them met the targeted annual percent growth.

Here we want to find the 99% confidence interval for the proportion of the portfolios meeting target goals managed by Kendra.

a) Formula of confidence interval for population proportion is

Where E is called margin of error

Formula of E is

b) Where,

n = sample size = 83

x = number of success

= x/n = 62/83 = 0.747

It is given that ; c = confidence level = 0.99

so that level of significance = \alpha = 1 - c = 1 - 0.99 = 0.01

this implies that /2 = 0.01/2 = 0.005

So we want to find   Z/2 such that

P(Z >Z/2 ) = 0.005

Therefore ,

P(Z < Z/2 }) = 1 - 0.005 = 0.995

The general excel command to find critical z value is

"=NORMSDIST(probability)"

Here probability = 0.995

So that critical  Z/2    is = "=NORMSINV(0.995)" = 2.5758

c)

So E = 0.123

Lower limit = - E = 0.747 - 0.123 = 0.624

Upper Limit = + E = 0.747 + 0.123 = 0.870

So that 99% confidence interval for population proportion P is (0.624, 0.870)

d) The mean of the sample proportion is same as sample proportion = 0.747

The standard deviation of the sample proportion is as follow

By Central Limit Theorem the distribution of the sample proportion is approximately normal with mean = 0.747 and the standard deviation = = 0.0477

So the sketch of the distribution with endponts is as follow:

e) Interpretation:

We find the a 99% confidence interval for the proportion of the portfolios meeting target goals managed by Kendra is (0.624, 0.870)

So if we are obtained confidence interval 100 times for the population proportion then 99 of the times the population proportion of the portfolios meeting target goals managed by Kendra lies in that intervals.