Calculate the percentage of refunds expected to exceed 1000

Question

QUESTION 1:

The data in the accompanying table gives the distances (in km) registered by a sample of company vehicles during a given week.

142	140	135	150	147	156	157	144
168	126	135	152	164	125	176	145
138	128	136	142	148	165	161	145
158	132	173	154	149	153	163	146
178	135	138	140	150	144	147	146

1.1 Using 6 classes of equal width, construct a less-than cumulative frequency distribution of the data in the above table. Let 120 be the lower limit of the initial class.

1.2 Using the information obtained in question 1.1, draw, to scale, a less-than ogive.

1.3 Use the less-than ogive to estimate the mid 70% range of the data.

1.4 Calculate the coefficient of variation and comment on the value obtained.

QUESTION 2:

An Inland Revenue Service (IRS) regional office in the United States has plotted a large sample of tax refund amounts and found that they form a bell-shaped distribution symmetrical about the central line. The average refund amount was found to be close to $750 with a standard deviation of $125. Most refund values are fairly close to the central line, suggesting a normal distribution.

The regional office is especially interested in large refund amounts. A refund greater than $1000 is considered large, and the office would like to know what percentage of current refunds, based on the normal distribution, are in excess of this amount.

A second question involves the office's concern with new withholding guidelines for taxpayers. It has been estimated that the average refund amount will rise by $120 after these guidelines go into effect, and the IRS office wonders how the percentage of large refunds will be affected.

2.1 Calculate the percentage of refunds expected to exceed $1000 under the current withholding guidelines.

2.2 Calculate the percentage increase in the refunds exceeding $1000 if the average refund increases by $120. Assume that the degree of variability in refund amounts remain unchanged when the average refund increases.

2.3 What would be the effect on the percentage of refunds over $1000 if the average refund amount actually drops by $70.

2.4 What change in the current average refund over $1000 will produce a 5% increase in the current percentage of refunds over $1000. (Assume no change in the degree of variability in refunds.)

QUESTION 3:

The table below gives two samples selected from 10 supermarkets of the weekly sales of a popular soft drink. The first sample gives the details for a normal shelf display of the product, while the second sample gives the details for an end-aisle shelf display. Assuming equal variances, establish, at the 5% level of significance, whether there is a statistically significant difference in the mean weekly sales for the two display locations.

Normal display	84	56	52	34	30	40	64	59	62	22
End Aisle Display	90	77	76	71	67	83	66	84	54	52

QUESTION 4:

An insurance company reported the number of policy holders who applied to surrender their endowment policies for each quarter during the last 4 years. The insurance has been communicating with the policy holders over the past four years to discourage them from surrendering their policy prior to maturity because of reduced payout on early surrender. The data is given in the table below.

Year	Apr-Jun	Jul-Sep	Oct-Dec	Jan-Mar
1	201	185	19	205
2	184	160	171	182
3	167	150	163	175
4	158	145	154	165

4.1 Present the above data in a time-series plot. Does the company's communication with policy holders appear to be working? Comment.

4.2 Use the ratio-to-moving average method and regression analysis to obtain seasonally adjusted trend estimates for each quarter of next year. Show details of your calculations. (Assume that the cyclical and irregular variations of the time-series are negligible.)

QUESTION 5:

An oil company is considering whether or not to bid for an offshore &Wing contract. The bid would cost $60m with a 65% chance of gaining the contract. The company may set up a new drilling operation or move its already existing operation, which has proved successful, to a new site. The probability of success and expected returns are as follows:

	New Operation		Move Existing Operation
Outcome	Probability	Expected Return ($m)	Probability	Expected Return ($m)
Success	0.7	80	0.8	70
Failure	0.3	20	0.2	30

Should company not bid or be unsuccessful in its bid, they can use the $60m to modernize their operations. This would result in a return of either 5% or 10% on the sum invested with probabilities 0.4 and 0.6 respectively.

With the aid of decision tree prepare a quantitative report advising the company on the optimal course of action.