For some reason people care about how acidic fresh water is. Two researchers conducted a study of

15 mountain lakes in the Southern Alps; any lake that has a pH greater than 6 would be classfied as

"non-acidic." Below is a table of the pH levels from the 15 lakes the researchers surveyed:

pH values of 15 Alpine Lakes

7.2 7.3 5.7

7.3 6.3 6.9

6.1 5.5 6.7

6.9 6.3 7.9

6.6 6.5 5.8

The standard deviation of the sample is given as: S = 0:672. At the 5% -level, does this sample

provide sufficient evidence to conclude that, on average, high mountain lakes in the Southern Alps are

non-acidic? State both null and alternative hypotheses, calculate and interpret your test statistic.

1. A Betting Game

You roll a die. If an odd number comes up, you lose. If you get a 6, you win $60. If it is an even number other than 6, you get to roll again. If you get a 6 the second time, you win $36. If not, you lose.

(a) Construct a probability model for the amount you win at this game. Explain briefly how you obtain the probabilities associated with the different amounts of winning.

(b) How much would you be willing to pay to play this game?

2. Chris Again!

A true-false test consists of 10 items.

(a) If Chris does not study at all and guesses each and every item in the test, describe the probability model for the number of correct guesses.

(b) What is the probability that Chris gets 80% or more for the test?

(c) If it is a 20 item true-false test, would you think it is easier or more difficulty for Chris to get 80% or more? Explain without performing any further calculation.

Let πx denote the proportion of households in some county with x cars. Suppose π1 = 0.32, π2 = 0.38, π3 = 0.1, π4 = 0.02, π5 = 0.01 and all the other households in the county have 0 cars.

(a) What is the average number of cars per household in this county? What is the corresponding variance?

(b) If a household is selected at random and X is its number of cars, draw the pmf of X.

(c) What is the expectation and what is the standard deviation of X?

(d) Compute the skewness of X.

You wish to test the following claim (HaHa) at a significance level of α=0.10α=0.10.

      Ho:μ=61.5Ho:μ=61.5
      Ha:μ<61.5Ha:μ<61.5

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=8n=8 with a mean of M=49.9M=49.9 and a standard deviation of SD=7.8SD=7.8.

What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value =

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

1. What happens to the validity of the EOQ model of ordering when the reorder point (ROP) exceeds the maximum inventory level during each cycle of replenishment? Use suitable sketches to support your answer. Limit your answer to two pages

2. For the normal distribution, “joint sufficiency condition” implies one needs only to know the mean and standard deviation of the data distribution to be able to apply it. With this as a fundamental principle, discuss the significance of the “Z-score”. Use suitable sketches to support your answer. Limit your answer to two pages

3. While solving linear programming problems involving two variables using the graphical method, will the optimal solution (assuming there is one) only be at corner points of the feasible solution space? Discuss with suitable sketches

BIG Corporation advertises that its light bulbs have a mean lifetime, u , of 3000 hours. Suppose that we have reason to doubt this claim and decide to do a statistical test of the claim. We choose a random sample of light bulbs manufactured by BIG and find that the mean lifetime for this sample is 2860 hours and that the sample standard deviation of the lifetimes is hours. Based on this information, answer the questions below. What are the null hypothesis (H0) u is less than, less than or equal to, greater than, greater than or equal to , not equal to, equal to 2860. 3000. 700? and the alternative hypothesis (H1) is ( same options above) 2860, 3000, 700? that should be used for the test? : is : is In the context of this test, what is a Type II error? A Type II error is rejecting, failing to reject the hypothesis that is when, in fact, is . Suppose that we decide to reject the null hypothesis. What sort of error might we be making? typeI or type II?

Selling Price        Living Area (Sq Feet)       No. Bathrooms No Bedrooms    Age (Years)

$240,000              2,022     2.5          3              20

$235,000              1,578     2              3              20

$500,075              3,400     3              3              20

$240,000              1,744     2.5          3              20

$270,000              2,560     2.5          3              20

$225,000              1,398     2.5          3              20

$280,000              2,494     2.5          3              20

$225,000              2,208     2.5          4              20

$248,220              2,550     2.5          3              20

$275,000              1,812     2.5          2              20

$137,000              1,290     1              2              20

$150,000              1,172     2              2              20

$649,000              4,128     3.5          3              20

$195,000              1,816     2.5          3              97

$373,200              2,628     2.5          4              20

$169,450              1,254     2.5          3              20

$144,200              1,660     1.5          4              20

$189,900              1,850     1.5          3              20

$166,000              1,258     2              3              20

$160,000              1,219     2              3              20

$327,355              1,850     2.5          3              20

$247,000              2,103     2.5          3              20

$318,000              1,806     2.5          3              20

$341,000              1,674     1.5          2              17

$288,650              2,242     2.5          3              20

$157,000              1,408     1.5          3              20

$449,000              3,457     2.5          3              21

$142,000              1,728     1.5          3              21

$389,000              2,354     2.5          3              21

$476,000              2,246     2.5          3              21

$249,230              1,902     2.5          2              21

$139,900              1,178     1              3              21

$301,900              2,896     3.5          4              21

$425,000              2,457     3              3              41

$121,000              936         1              3              50

$150,000              934         1              2              21

$138,000              1,279     1              3              21

$199,900              1,888     2              3              26

$145,000              1,686     1.5          4              21

$465,000              2,310     3              2              21

$158,000              1,200     1.5          3              21

1.             Develop a multiple linear regression model to predict the price of a house using the square feet of living area, number of bedrooms, number of bathrooms, and age as the predictor variables.

1.             Write the reqression equation.
2.             Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.
3.              Discuss the statistical significance of the coefficient for each independent variable using the appropriate regression statistics at a 95% level of confidence.
4.             Discuss the statistical significance of the coefficient for each independent variable using the appropriate regression statistics at a 90% level of confidence.

2.             Develop a multiple linear regression model to predict the price of a house using the square feet of living area, number of bedrooms, and number of bathrooms as the predictor variables.

1.             Write the reqression equation.
2.             Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.
3.              Discuss the statistical significance of the coefficient for each independent variable using the appropriate regression statistics at a 95% level of confidence.
4.             Interpret the coefficient for each independent variable.
5.             What percentage of the observed variation in housing prices is explained by the model?
6.              Determine the prediction interval of a house with 3,000 square feet of living area, 3 bedrooms, and 2.5 bathrooms, and comment on the prediction interval.

3.             In Case Study No.1 you ran a simple linear regression model for predicting the price of a house based upon its living area in square feet using a 95% level of confidence. Rerun that simple linear regression model for predicting the price of a house based upon its living area in square feet using the Case study 2 data using a 95% level of confidence. Compare the regression statistics for this simple linear regression model with the statistics for the preceding multiple linear regression model using square feet of living area, number of bedrooms, and number of bathrooms as the predictor variables. Which model is the preferred model at a 95% level of confidence? Use the Significance F values, p-values for the independent variables, R-squared or Adjusted R-squared values (as appropriate), and standard errors to explain your selection.

Prepare a single Microsoft Excel file to document your regression analyses. Prepare a single Microsoft Word document that outlines your responses for each portion of the case study.

Answer anyone question

Question 1:  Fama and French (1993) have included a Size factor in their 3-factor model. Survey the most recent literature and discuss whether the Size factor is still present in financial markets (in the US and elsewhere). Using the Methodology of Fama and MacBeth (1973), test and discussion possible economic reasons why Size should be a priced factor. Requires a large literature base and a simple regression analysis. If you are fully prepared, I hope to write this question.

Question 2: Carhart (1997) advocated momentum as another factor. Survey the most recent literature and discuss whether the momentum factor is still present in financial markets (in the US and elsewhere). Using the methodology of Fama and MacBeth (1973), test and discuss possible Economic reasons why Size should be a priced factor. Requires a large literature base and a simple regression analysis.

n a research project, researchers collected demographic and health data from a sample of elderly residents in the community. To examine any possible gender differences in their sample, they want to see if the females and the males differ significantly on the education level (number of years of formal schooling). The researchers are not predicting any direction in the possible gender differences so the hypotheses should be non-directional. They would like to run a two-tailed test with α = .10.  

n a research project, researchers collected demographic and health data from a sample of elderly residents in the community. To examine any possible gender differences in their sample, they want to see if the females and the males differ significantly on the education level (number of years of formal schooling). The researchers are not predicting any direction in the possible gender differences so the hypotheses should be non-directional. They would like to run a two-tailed test with α = .10.  

1. Calculate estimated variance for population for population 1 (S1^1) and (S2^2)

2.Calculate the pooled variance (Spooled2) from the two population variances

3.Use the pooled variance (from question f above) to calculate the variance for sampling distribution 1 (SM12) and the variance for sampling distribution 2 (SM22?

4.Calculate standard deviation (Sdiffmean)of the comparison distribution

5. calculate t statistic and critical t values

A farmer who specialises in the production of carpet wool where the sheep are shorn twice per year is seeking a 75-mm-length clip from his Tukidale sheep. He believes that the proportion of sheep at each clip meeting this standard varies according to average rainfall during the six-month growing period and whether additional hand feeding of high protein sheep nits occurs during the period (because of a shortage of grass cover in the paddocks). Hand feeding is measured as 1 and no hand feeding as 0. Showing ALL formulas and working; a) Predict the proportion at 75mm if the rainfall is 180 mm and there is no hand feeding, a construct a 95% confidence interval estimate and 95% prediction interval.

e) Calculate the coefficients of partial determination and interpret their meaning.

f) Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model.

Last week, 108 cars received parking violations in the green parking lot. Of these, 27 had unpaid parking tickets from a previous violation. Assuming that last week was a random sample of all parking violators, find the width of the 90 percent confidence interval for the percentage (correct to four decimal places) of parking violators that have prior unpaid parking tickets.

Suppose that a population of brakes supplied has a mean stopping distance, when the brake is applied fully to a vehicle traveling at 75 mph is 268 feet. Population standard deviation is 20 feet. Suppose that you take a sample of n brakes to test and if the average stopping distance is less than or equal to a critical value, you accept the lot. If it is more than the critical value, you reject the lot. You want an alpha of 0.03. Further, we want to reject lots with a population mean stopping distance of 282 ft., we want to reject with a probability of 0.9. (Note that it is one sided, since we would not get worried if the vehicle stops at a distance shorter than the average time.)

1. Draw a diagram and mark the two distributions, alpha, beta and CV.

2. What is the correct sample size that can achieve this?

3. What is the critical value?

Question:

Discuss the reasons for using Bayesian analysis when faced with uncertainty in

making decisions.

Discussion Requirements:

How would you describe Bayesian Theorem?

Describe the assumptions of Bayesian analysis.

Provide the example of problem where one can use Bayesian analysis in Big Data Analytics.

Describe the the problems with Bayesian analysis.

Suppose a researcher is collecting data about the amount of soda Americans consume in one day. If the researcher was to gather two separate representative samples from the population, differences may exist between these samples. These differences may be attributed to what?