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QNM222 Business Statistics01

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  • Course Code: QNM222
  • University: Seneca College
  • Country: Canada

Question:

Question 1

The Rutter Nursery Company packages its mulch in 50 kg bags. From a long history, the shipping department reports that the distribution of the bag masses follows the normal distribution and the standard deviation of this process is 3 kg per bag. At the end of each day, Jeff Rutter, the shipping manager, selects 10 bags and computes the mean mass of the sample. The sample mean was found to be 48.18 kg.

a)Can Mr. Rutter conclude that the mean mass of the bags is less than 50 kg? Use the 1% significance level.

b)Calculate the p-value.

Question 2

In recent years, the interest rates on home mortgages have declined to less than 6%. However, a recent study shows that the rate charged on credit card debt is more than 14%. A sample of 10 credit cards showed that the mean rate charged is 15.64% with a standard deviation of 1.561%. At 1% level of significance, is it reasonable to conclude the mean rate charged is greater than 14%?

Question 3

A shopping centre wants to examine the amount of space required for parking. Studies indicated that 50% of staff and shoppers use public transportation. A survey of 1002 was taken, and 483 responded that they used public transportation. At 5% level of significance, is it reasonable to conclude that the survey results indicate a change?

 

Answer:

Answer 1

Hypothesized mean mass of all bags (µ) = 50 kg

Sample mean mass of bags  = 48.18 kg

Standard deviation of the sample masses  = 3 kg

Sample size (n) = 10 bags

Degrees of freedom = 9

Type of test = Upper tail test

  • To test at 1% level of significance, whether the mean mass of the bags is less than 50 kg, the following hypothesis can be framed:

The test statistic to test the above hypothesis can be given as follows:

The critical value of t with 9 degrees of freedom at 1% level of significance  = 2.8214. Thus,  < . Hence, null hypothesis () is accepted. Thus, Mr. Rutter cannot conclude that the mean mass of the bags is less than 50 kg.

  • The p-value is given by
 

Answer 2

Hypothesized mean rate of credit card debt (µ) = 14

Sample mean rate of credit card deb  = 15.64

Standard deviation of the sample rates  = 1.561

Sample size (n) = 10

Degrees of freedom = 9

Type of test = Lower tail test

 at 1% level of significance = – 2.8214

To test at 1% level of significance, whether the mean rate of credit card debt is greater than 14%, the following hypothesis can be framed:

The test statistic to test the above hypothesis can be given as follows:

Thus,  > . Hence, null hypothesis () is accepted. Thus, it is reasonable to conclude that the mean rate charged is greater than 14%.

Answer 3

Population proportion of staffs and shoppers using public transportation (P) = 0.5

Sample size = 1002

Number of people using public transportation = 483

Sample proportion =

Type of test = Two tailed test.

 at 5% level of significance = 1.96

To test whether the proportion of staffs and shoppers using public transportation have changed, the following null and alternate hypothesis can be framed:

The test statistic to test the above hypothesis can be given as follows:

Thus,  < . Hence, null hypothesis () is accepted. Thus, it is reasonable to conclude that the survey results do not indicate a change.

Answer 4

  • The scatter diagram showing the relationship between the age and the selling price is given in figure 4.1

Figure 4.1: Scatter Diagram

  • Here, mean of age =

Average selling price

The correlation coefficient between age of the car and its selling price is:

Thus, there is a moderately negative relationship between age and selling price. This result is not surprising as with the increase in age of a car, the functionality of the car will decrease and thus, the selling price will also decrease.

  • The coefficient of determination between age and selling price is . This measure indicates that 29 percent of the variability in the selling price of the car can be explained by its age.
  • To test whether there is a linear relationship between the variables age and its selling price, the following hypothesis can be framed:

Here,  is the regression coefficient. From the regression analysis, it can be seen that,  and the result is significant at 5% level of significance. Thus, there exists a linear relationship between age and selling price.

 

Table 4.1: Regression Coefficients

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

11.58

2.36

4.91

0.00

6.32

16.83

Age (years) [X]

- 0.51

0.25

-2.03

0.07

-1.08

0.05

  • The regression equation that can be given to estimate the selling price of the car based on its age is
  • Here, the value of b (slope of the regression equation) is – 0.51. This indicates that with the increase in age of the car by 1 year, the selling price of the car decreases by 0.51.
  • The estimated selling price of a 10-year-old car is:

Therefore, the required estimated selling price of a 10-year old car is $6,440

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