Largest private companies. IPOs—initial public offerings of stock—create billions of dollars of new wealth for owners, managers, and employees of companies that were previously privately owned. Nevertheless, hundreds of large and thousands of small companies remain privately owned. The revenues of a random sample of 15 firms from Forbes 216 Largest Private Companies list are given in the table below

a. Describe the population from which the random sample was drawn.

b. Use a 98% confidence interval to estimate the mean revenue of the population of companies in question

c. Interpret your confidence interval in the context of the problem

d. What characteristic must the population possess to ensure the appropriateness of the estimation procedure used in part b?

e. Suppose Forbes reports that the true mean revenue of the 216 companies on the list is $5.0 billion. Is the claim believable?

Largest private companies. IPOs—initial public offerings of stock—create billions of dollars of new wealth for owners, managers, and employees of companies that were previously privately owned. The revenues of a random sample of 15 firms from Forbes 216 Largest Private Companies list are given in the table above.

a)

The population is the totality of all individuals about which we want to collect information. The random variable sample contains 15 firms from the Forbes 216 Largest Private Companies List. Hence the population is all 216 firms on the firms on the Forbes 216 Largest Private Companies list.

b)

Sample size 15

Confidence coefficient 98%=0.98

Let x be the amount of revenues of the companies in the list.

Mean:

There are 15 data values

$\begin{array}{c}\sum _{\mathrm{i}=1}^{15}{\mathrm{x}}_{\mathrm{i}}=12.4+31+\mathrm{...}+5.1+2.3\\ =111.5\end{array}$

The mean is

$\begin{array}{c}\overline{\mathrm{x}}=\frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{i}}}}{\mathrm{n}}\\ =\frac{111.5}{15}\\ =7.4333\end{array}$

Standard deviation

$\begin{array}{c}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{i}}^2={12.4}^2+{31}^2+\mathrm{...}+{5.1}^2+{2.3}^2\\ =1586.53\end{array}$

If s be the sample variance then

$\begin{array}{c}{\mathrm{s}}^2=\frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{i}}^2-\frac{{\displaystyle \left(\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{i}}\right)}}{\mathrm{n}}}}{\mathrm{n}-1}\\ =\frac{1586.53-\frac{{111.5}^2}{15}}{15-1}\\ =54.1224\end{array}$

Now, sample standard deviation is

$\begin{array}{c}\mathrm{s}=\sqrt{{\mathrm{s}}^2}\\ =\sqrt{54.1224}\\ =7.3568\end{array}$

Critical value

The critical value of t at 15-1=14 degree of freedom and at $\mathrm{\alpha }=2\mathrm{\%}$is${\mathrm{t}}_{\frac{\mathrm{\alpha }}{2}}=2.624$

The margin of error

$\begin{array}{c}\mathrm{E}={\mathrm{t}}_{\frac{\mathrm{\alpha }}{2}}\times \frac{\mathrm{s}}{\sqrt{\mathrm{n}}}\\ =2.624\times \frac{7.3568}{\sqrt{15}}\\ =4.9843\end{array}$

Confidence interval

98% lower bound$\overline{\mathrm{x}}-\mathrm{E}=7.4333-4.9843=2.4490$

98% upper bound$\overline{\mathrm{x}}+\mathrm{E}=7.4333+4.9843=12.4176$

So, 98% confidence interval to estimate the mean revenue of the population of companies is$(2.4490,12.4176)$.

c)

One is 98% confident that true mean revenue that the true mean revenue is between 2.4490 billion and 12.4176 billion dollars.

d)

It is required that the following characteristics must be required to satisfy to ensure the estimation process used in part b.

1.random sample

2. the sample size must be greater than 30 or the distribution must follow normal distribution

Here from the description it can be assumed that the sample is a random sample . Now the sample size is 15 which indicates the normality condition must be satisfied.

So, the population distribution needs to be approximately normal.

e)

The 98% confidence interval to estimate the mean revenue of the population of companies is $(2.4490,12.4176)$. Since $5.0 billion contains in the interval it is believable that The claim that Forbes reports that the true mean revenue of the 216 companies on the list is $5.0 billion.