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Statistics for Business

Unit: STA101 – Statistics for Business

Each of the following question worth 10 marks, please answer all of them. If needed please use Harvard referencing style. There is no word limit, but it is necessary that you provide answers with explanations.

Question 1: 

A sample of eight observations of variables x (years of experience) and y (salary in $1,000s) is shown below:

x
5
3
7
9
2
4
6
8
y
20
23
15
11
27
21
17
14



  1. Calculate and interpret the covariance between x and y.
  1. Give a possible reason that the covariance is negative.

  1. Calculate the coefficient of correlation, and comment on the relationship between x and y.

  1. Give a possible reason that the correlation is negative.

Question 2: 

The L. L. Bean catalog department that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 3 minutes.

  1. What is the value of l, the parameter of the exponential distribution in this situation?
  2. What proportion of customers having to hold more than 1.5 minutes will hang up before placing an order?
  3. Find the waiting time at which only 10% of the customers will continue to hold.
  4. What is the probability that a randomly selected caller is placed on hold for 3 to 6 minutes?

Question 3: 

A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H0:  = 950 hours vs. H1:  950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.

  1. Calculateb, the probability of a Type II error when m = 1000 and a = 0.10.
  2. Calculate the power of the test when m = 1000 and a = 0.10.
  3. Interpret the meaning of the power of the test.
  4. Review the results of the previous questions. What is the effect of increasing the sample size on the value of b?

Question 4

A production filling operation has a historical standard deviation of 6 ounces. When in proper adjustment, the mean filling weight for the production process is 47 ounces. A quality control inspector periodically selects at random 36 containers and uses the sample mean filling weight to see if the process is in proper adjustment. Using a standardized test statistic, test the hypothesis at the 5% level of significance if the samples mean filling weight is 48.6 ounces.

Answer

Question 1: 

A sample of eight observations of variables x (years of experience) and y (salary in $1,000s) is shown below:


x53792468
y2023151127211714



  1. Calculate and interpret the covariance between x and y.
The covariance is -12.71. This indicates that as value of x increases, the value of y decreases. Further the linear relationship is of degree 12.71, meaning that a unit of change in x leads to opposite 12.71 units change in y.
  1. Give a possible reason that the covariance is negative.
Typically, as years of experience increase, the salary must increase. However, a possible reason for negative covariance may be that the samples are from different industries where salary levels are not consistent. Hence, while one observation may be from a highly paying industry, another may be from an industry that does not pay so well.
  1. Calculate the coefficient of correlation, and comment on the relationship between x and y.
The coefficient of correlation is -0.99. The correlation normalizes the covariance value. Hence, given any two variables value, the increase in x will be followed by decrease in y.
  1. Give a possible reason that the correlation is negative.
The reason will be similar as quoted in part b., that is, the samples are from different industries. 

Question 2: 

The L. L. Bean catalog department that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 3 minutes.

  1. What is the value of l, the parameter of the exponential distribution in this situation?

l-1=3

l=1/3 = 0.333

l= 0.333

  1. What proportion of customers having to hold more than 1.5 minutes will hang up before placing an order?

1-0.6065 = 0.3935 or 39.35% of the customers

  1. Find the waiting time at which only 10% of the customers will continue to hold.

Approximately 6.9 minutes

  1. What is the probability that a randomly selected caller is placed on hold for 3 to 6 minutes?

X(3) = 0.36788

X(6) = 0.13534

X(6)-X(3) = 0.23254

Or, 23.25%

Question 3: 

A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H0:  = 950 hours vs. H1:  950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.

  1. Calculateb, the probability of a Type II error when m = 1000 and a = 0.10.

m = 1000, a = 0.10, n = 25

Using this formula, we calculate beta = 0.0019 or 0.19%

  1. Calculate the power of the test when m = 1000 and a = 0.10.

Power is 1

  1. Interpret the meaning of the power of the test.

The main effect is of sample size while calculating power. Power indicates the probability of rejecting null hypothesis when it is false. Hence, power indicates the probability of making correct decision through testing, that is, rejecting null when it is false. Or, not making type II error.  

  1. Review the results of the previous questions. What is the effect of increasing the sample size on the value of b?

The beta will reduce as sample size increases. Since Type II error probability is declining, power of the etst will keep increasing as sample size increases.

Question 4

A production filling operation has a historical standard deviation of 6 ounces. When in proper adjustment, the mean filling weight for the production process is 47 ounces. A quality control inspector periodically selects at random 36 containers and uses the sample mean filling weight to see if the process is in proper adjustment. Using a standardized test statistic, test the hypothesis at the 5% level of significance if the samples mean filling weight is 48.6 ounces

σ = 6

m = 48.6

N = 36

a = 0.05

Sample mean = 48.6

H0: m = 47

H1: m ≠ 47

T stat = (48.6-47)/(6/sqrt36) = 1.6/1 = 1.6

Df = 35

p-value = 0.118588

The result is not significant as p-value<0.5. Hence, we reject the null hypothesis. Mean is not 47 ounces.

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