Excess revenue (total revenue minus operating expenditures) in the nonprofit sector are normally distributed with a mean of $1.5 million and a standard deviation of $1 million.

(a) What is the probability that a randomly selected nonprofit has negative excess revenues?

(b) What is the probability that a randomly selected nonprofit has excess revenue between $1 million and $2 million?

(c) If 10% of nonprofits are expected to exceed a certain excess revenue level, what is that revenue level?

Normal (or Gaussian) distributions are widely used in practice because many sets of observations follow a bell-shaped curve. In statistics, the normal distribution is one of the main assumptions in statistical inferences, such as confidence intervals and hypothesis tests.

After conducting some basic searches using scholarly articles, explain how normal distributions are used in business analytics. Your findings must include:

1. The problem background
2. Why a normal distribution is used
3. How is a normal distribution used?
4. What are the results?

Suppose you have just received a shipment of 27 modems. Although you​ don't know​ this, 3 of the modems are defective. To determine whether you will accept the​ shipment, you randomly select 8 modems and test them. If all 8 modems​ work, you accept the shipment.​ Otherwise, the shipment is rejected. What is the probability of accepting the​ shipment?

A city manager is considering three strategies for a $1,000 investment. The probable returns are estimated as follows: • Strategy 1: A profit of $5, 000 with a probability of 0.20 and a loss of $1, 000 with a probability of 0.80.

• Strategy 2: A profit of $2, 000 with a probability of 0.40, a profit of $500 with a probability of 0.30 and a loss of $1, 000 with a probability of 0.30.

• Strategy 3: A certain profit of $400.

(a) Which strategy has the highest expected profit?

(b) If the city manager is going to pick only 1 strategy, which of the three strategies would you definitely advise against? Provide specific (numeric) details to support your answer.

Test the claim that the proportion of men who own cats is larger than 20% at the .05 significance level.

The null and alternative hypothesis would be:

H0:p=0.2H0:p=0.2
H1:p≠0.2H1:p≠0.2

H0:p=0.2H0:p=0.2
H1:p>0.2H1:p>0.2

H0:μ=0.2H0:μ=0.2
H1:μ≠0.2H1:μ≠0.2

H0:μ=0.2H0:μ=0.2
H1:μ<0.2H1:μ<0.2

H0:p=0.2H0:p=0.2
H1:p<0.2H1:p<0.2

H0:μ=0.2H0:μ=0.2
H1:μ>0.2H1:μ>0.2

The test is:

right-tailed

two-tailed

left-tailed

Based on a sample of 30 people, 29% owned cats

The test statistic is:  (to 2 decimals)

The critical value is:  (to 2 decimals)

Based on this we:

• Reject the null hypothesis
• Fail to reject the null hypothesis

As shown in Figure 02, an urn contains 12 red balls and 4 green balls. The red balls are numbered from 1 to 12, and the green balls are numbered from 1 to 4. One ball is randomly drawn from the urn. Which of the following answers is correct? (Let: R = red; G = green; and E = even.)

1. P(G ∪ R) = 0.000.

2. P(R|E) = 0.375.

3. P(G ∪ E) = 0.625.

4. P(G|R) = 0.500.

Please provide a walkthrough explanation on each answer given.

For 300 trading​ days, the daily closing price of a stock​ (in $) is well modeled by a Normal model with mean ​$197.49197.49 and standard deviation ​$7.147.14. According to this​ model, what is the probability that on a randomly selected day in this period the stock price closed as follows. ​

a) above ​$211.77211.77​?

​b) below ​$204.63204.63​? ​

c) between ​$183.21183.21 and ​$211.77211.77​?

​d) Which would be more​ unusual, a day on which the stock price closed above ​$210210 or below ​$190190​?

In a debate on altering the traffic system in the city centre, measurement of a number of cars per minutes were taken at two intersections during the hours between 07h00 and 08h00 (when the roads were most busy). The results are shown in the table below:

1. Average
2. Median and the mode
3. Standard deviation
4. Interquartile range
5. Co-efficient of variation

Discuss appropriate data representations

Find the percent of the area to the left of

z = −2.35.

A psychologist wants to know whether wives and husbands who both serve in a foreign war have similar levels of satisfaction in their marriage. To test this, six married couples currently serving in a foreign war were asked how satisfied they are with their spouse on a 7-point scale ranging from 1 (not satisfied at all) to 7 (very satisfied). The following are the responses from husband and wife pairs.

(a) Test whether or not mean ratings differ at a 0.05 level of significance. State the value of the test statistic. (Round your answer to three decimal places.)

(b) Compute effect size using eta-squared. (Round your answer to two decimal places.)

. Taxation and comparative statics with small changes:

a. Given that ??/?? represents the tax incidence on consumers in equation, (??/??= ?/(?−?))what is the expression for the tax incidence on firms in terms of η & ε. What does the sum of the tax incidence on consumers and firms equal? In no more than two sentences, explain why they must sum to this number.

b. Now consider another scenario, the imposition of a specific tax (?) that is paid directly to the

government by consumers rather than by firms as in part (a). Under such a tax, the equilibrium

condition is now: ?(?∗(?) + ?) = ?(?∗(?)). This condition sets a general demand function equal to a

general supply function, shows that the equilibrium price P* is an implicit function of ?, and makes

explicit that the amount now paid for the good by consumers with the tax is ?∗ + ?. Use this equilibrium

condition to find ??/??.

c. In part (a) P* represents the price consumers pay. What does P* represent in part (b)?

d. Should it be surprising that your answers to parts (a) and (b) represent the same tax incidence for firms?

Support your answer in no more than three sentences.

Suppose that Y has the gamma distribution with parameters a ( shape ) = 2 and b (scale) = 2. Use R to plot the probability density, and determine the shape or skewness for the gamma distribution.

Use R code

Describe the steps necessary to calculate the variance of a sample. Why is it important to understand variance?

Discuss a situation in which you have used the variance of a sample to help solve a problem.