Statistics - QAB 105 Quantitative Analysis For Business - Assessment Answer

Statistics Assignment

Assignment Task1

a) Make a scatterplot for these data.

b) Describe the association between age and price of a used Corolla. Do you think a linear model is appropriate?

c) Computer software says that r 2 = 0.894. What is the correlation between age and price?Explain the meaning of r 2 in this context.

d) Why doesn’t this model explain 100% of the variability in the price of a used Corolla?

e) P Given ^ = 12319.6 the estimated – 924A, linear where model ^

P is predicted for the relationship price and A between is age of a car. car’s Answer age and the its following price is:

questions:

Explain the meaning of the slope of the line, and the y-intercept of the line.

If you want to sell a 7-year-old Corolla, what price seems appropriate?

You have a chance to buy one of two cars. They are about the same age and appear to be in equally good condition. Would you rather buy the one with a positive residual or a negative residual? Explain.

You see a “For Sale” sign on a 10-year-old stating the asking price as $1500. What isthe residual?

Would this regression model be useful in establishing a fair price for a 20-year-oldcar? Explain

Assignment Task2

If Tennant Creek Town’s daily water demand is approximately normally distributed with a mean of 5 ml and a standard deviation of 1.25ml:

a) Estimate the number of days in a (365 day) year on which daily consumption is:50% or more greater than the mean.

ii. within two standard deviation of the mean.

iii. below the first quartile level of demand.

b) If the water supply authority decides to save money by setting supply capacity to a level adequate to satisfy daily demand on 95% of all days at what level should capacity be set?

Assignment Task3

An executive of a new telephone company wants to know whether the average length of evening long-distance telephone calls in a metropolitan area still equals 18.1 minutes, as it did in the past. A simple random sample of 25 evening calls is to be used to find the answer at a significance level of ?=0.05. After taking a sample of n = 25, the statistician finds a sample mean duration of calls of 17.2 minutes and sample variance of 4 minutes squared.

a) Formulate the null and alternative hypothesis.

b) What is the critical value and state the rejection rule?

c) What is the value of the test statistics?

d) What is the p-value for the test?

e) What is your conclusion?

Solution1:

Scatter plot is obtained below:

We observe a decreasing trend from the above scatter plot, the relationship between the age and Price is related in a negative linear sense. It is seen that the points closely follow along a straight line, indicating that the assumption of linearity between the two variables appears to be reasonable.

The coefficient of determination, r², is 0.894 , the value of Correlation value = ?0.894 = -0.945. R-square value is 894 which indicates that there is approximately 89.4% of the total variation in the dependent variable is explained by the independent variables.

This model does not explain 100% of the variability in the price since the relationship is not perfect. Other factors such as options, mileage and condition explain the rest of the variability in price.

Explain the meaning of the slope of the line, and the y-intercept of the line.The slope of a line symbolizes the general direction in which a line points towards. The slope of a line is obtained by dividing the difference of the y-coordinates with the difference of the x-coordinates.y-intercept is a point in the graph where the x is zero.

If you want to sell a 7-year-old Corolla, what price seems appropriate?You have a chance to buy one of two cars. They are about the same age and appear to be in equally good condition. Would you rather buy the one with a positive residual or a negative residual? Explain.

When the residual is a positive value, the observed values are higher that the predicted values. This implies that most of the time, the average price of the car was higher with age. When a car is of good quality or has a low mileage, people tend to sell it at a higher price. The car with the negative residual implies that most of the time it has been on sale the price has been lower than its age. This may be due to poor quality build of the car or high mileage.

The best choice therefore would be to purchase the car that has a positive residual.

You see a “For Sale” sign on a 10-year-old stating the asking price as $1500. What is the residual?

Would this regression model be useful in establishing a fair price for a 20-year-old car? Explain

P=12319.6 - 924(20) =-6160.4

This regression model would not be useful in establishing the price for a 20-year-old car. It is not possible to sell a car at a negative value.

Solution2

Estimate the number of days in a (365 day) year on which daily consumption is:50% or more greater than the mean.

within two standard deviation of the mean.

below the first quartile level of demand.

If the water supply authority decides to save money by setting supply capacity to a level adequate to satisfy daily demand on 95% of all days at what level should capacity be set?

From the Normal Table,

The supply authority should set the supply capacity to 7.05 ml so as to satisfy the daily demand on 95% of all days.

Solution3

The information given in the problem can be represented with the following notations:

Sample size: n = 25

Sample mean: = 17.2 minutes

Sample variance: s2 = 4 minutes

Population mean: m= 18.1 minutes

Claim:the average length of evening long-distance telephone calls in a metropolitan area still equals 18.1 minutes

Null Hypothesis:

H₀: m= 18.1

That is H₀: The average length of evening long-distance telephone calls in a metropolitan area is equal to 18.1 minutes

Alternative Hypothesis:

H₁: m? 18.1 (two-tail test)

That is H₁: The average length of evening long-distance telephone calls in a metropolitan area is different from 18.1 minutes

Level of Significance: ? = 0.05

Critical value of tat ? = 0.05 level of significance (two tail-tests) is obtained from the t-distribution table and is given below:

t_{?/2, n-1.df}= t_{0.025, 24.df}= 2.064 (by referring t-distribution table)

Rejection rule:

Reject H0 if t> +2.064 or if t< ?2.064

Otherwise do not reject H0

The decision rule is illustrated in Figure

Test Statistic:

t = - 2.25

Thus, the value of test statistic ist = -2.25

P value= 2*P(t<-2.25) = 0.0339

Conclusion:

The test statistic falls in the rejection region, so we reject the null hypothesis at the 5 percent level of significance and conclude that the average length of evening long-distance telephone calls in a metropolitan area is different from 18.1 minutes