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Solving Various Statistics Questions

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1. A data set with 7 observations yielded the following. Use the simple linear regression model.
Greek letter Capital Sigma X =21.57

Greek letter Capital Sigma X2 =68.31
Greek letter Capital SigmaY =188.9

Greek letter Capital Sigma Y2 =5,140.23
Greek letter Capital Sigma XY =590.83
Calculate the Correlation coefficient, Coefficient of determination and SSE, and the Standard Error of Estimate.

2. A set of final examination grades in a calculus course was found to be normally distributed with a mean of 77.2 and standard deviation of 5. Only 2.5% of the students taking the test scored higher than what grade?

3. An apple juice producer buys all his apples from a conglomerate of apple growers in one northwest state. The amount of juice squeezed from each of these apples is approximately normally distributed with a mean of 2.3 ounces and a standard deviation of 0.15 ounce.
Between what two values (in ounces) symmetrically distributed around the population mean will 90% of the apples fall?

4. Consider the following partial computer output for a multiple regression model.
Predictor Coefficient Standard Deviation

Constant 41.225 6.380
X1 1.081 1.353
X2 -18.404 4.547

Analysis of Variance
Source DF SS
Regression 2 2270.11
Error 26 3585.75
What is the number of Observations in the sample? Write the least squares regression (prediction) equation. Test the usefulness of variable x_2 in the model at alpha =.05. Calculate the t statistic and state your conclusions.

5. A small town has a population of 20,000 people. Among these 2,000 regularly visit a popular local bar. A sample of 225 people who visit the bar is surveyed for their annual expenditures in the bar. It is found that on average each person who regularly visits the bar spends about $2500 per year in the bar with a standard deviation of $450. Construct a 99 percent confidence interval around the mean annual expenditure in the bar.

6. A data set with 7 observations yielded the following. Use the simple linear regression model.
Greek letter Capital Sigma X =21.57
Greek letter Capital Sigma X^2 =68.31
Greek letter Capital Sigma Y =188.9
Greek letter Capital Sigma Y^2 =5,140.23
Greek letter Capital Sigma XY =590.83
Write the Regression Equation showing Intercept and slope and test the significance of slope at 1% significance level.

7.Test H_0: pi_1- pi_2 <=0.01, HA: pi_1- pi_2 > .01 at (alpha)(lamba)(pi)(eta)(alpha)=0.05 and 0.10 where p_1=.08, p_2=.035, n_1 = 200, n_2 = 400. Indicate which test you are performing; show the test statistic and the critical values and mention whether one-tailed or two-tailed.
Note that pi stands for the Greek letter representing population proportion.

8. A human resource manager is interested in whether absences occur during the week with equal frequency. The manager took a random sample of 100 absences and created the following table:

Monday 28
Tuesday 20
Wednesday 12
Thursday 18
Friday 22

At a significance level of alpha = .05 test the Null that the probabilities of absences are the same for all five days.

9.Test H_0: mu=42 versus HA: mu is not equal to 42 when X-bar = 42.6, s=1.2 and n=16 at alpha=.01 and .05. Assume that the population from which the sample is selected is normally distributed. Indicate which test you are performing; show the test statistic and the critical values and mention whether one-tailed or two-tailed. (Note mu is the Greek letter for Mean)

10. The weight of a product is normally distributed with a standard deviation of .5 ounces. What should the average weight be if the production manager wants no more than 10% of the products to weigh more than 6 ounces?

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The solution gives detailed steps on solving various questions on fundemantals of statistics. The topic includes correlation coefficient, coefficient of determination, SSE, t statistic so on. All the formula and calcuations are shown and explained.

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Solving various questions on fundemantals of statistics

12. Which of the following numbers could be the probability of an event?

1.5 , 1/2 , 3/4 , 2/3 , 0 , -1/4

34. For some diseases, such as sickle-cell anemia, an individual will get the disease only if he or she receives both recessive alleles. This is not always the case. For example, huntington's disease only requires one dominant gene for an individual to contract the disease. Suppose that a husband and wife, who both have a dominant Huntington's disease allele (S) and a normal recessive allele (s), decide to have a child.

a) List the possible genotypes of their offspring.
b) What is the probability that the offspring will not have Huntington's disease? In other words, what is the probability that the offspring will have genotype ss? interpret this probability.
c) What is the probability that the offspring will have Huntington's disease?

40. Which of the assignments of probabilities should be used if the coin is known to be fair?

Sample Spaces
Assignments HH HT TH TT
A 1/4 1/4 1/4 1/4
B 0 0 0 1
C 3/16 5/16 5/16 3/16
D 1/2 1/2 -1/2 1/2
E 1/4 1/4 1/4 1/8
F 1/9 2/9 2/9 4/9

48. Determine whether the probabilities on the following page are computed using classical methods, empirical methods, or subjective methods.

a) The probability of having eight girls in an eight-child family is 0.390625%
b) On the basis of a survey of 1000 families with eight children, the probability of a family having eight girls is 0.54%
c) According to a sports analyst, the probability that the Chicago Bears will win their next game is about 30%
d) On the basis of clinical trials, the probability of efficacy of a new drug is 75%

26. The following probability model shows the distribution of doctoral degrees from U.S. universities in 2009 by area of study.

Area of Study Probability
Engineering 0.154
Physical Sciences 0.087
Life Sciences 0.203
Mathematics 0.031
Computer sciences 0.033
Social sciences 0.168
Humanities 0.094
Education 0.132
Professional and other fields 0.056
Health 0.042
Source: US National Science Foundation

a) Verify that this is a probability model.
b) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 studied physical science or life science? Interpret this probability.
c) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 studied physical science, life science, mathematics, or computer science? Interpret this probability.
d) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 did not study mathematics? Interpret this probability.
e) Are doctoral degrees in mathematics unusual? Does this result surprise you?

32. A standard deck of cards contains 52 cards. One card is randomly selected from the deck.
a) Compute the probability of randomly selecting a two or three from a deck of cards.
b) Compute the probability of randomly selecting a two or three or four from a deck of cards.
c) Compute the probability of randomly selecting a two or club from a deck of cards.

8. Determine whether the events E and F are independent or dependent. Justify your answer.
a) E: The battery in your cell phone is dead.
F: The batteries in your calculator are dead.
b) E: Your favorite color is blue.
F: Your friend's favorite hobby is fishing.
c) E: You are late for school.
F: Your car runs out of gas.

18. The probability that a randomly selected 40-year-old female will live to be 41 years old is 0.99855 according to the National Vital Statistics Report, Vol. 56, No. 9.
a) What is the probability that two randomly selected 40-year-old females will live to be 41 years old?
b) What is the probability that five randomly selected 40-year old females will live to be 41 years old?
c) What is the probability that at least one of five randomly selected 40-year-old females will not live to be 41 years old? Would it be unusual if at least one of five randomly selected 40-year-old females did not live to be 41 years old?

34. In how many ways can 15 students be lined up?

46. In how many ways can the top 2 horses finish in a 10-horse race?

52. How many different random samples of size 7 can be obtained from a population whose size is 100?

56. How many distinguishable DNA sequences can be formed using one A, four Cs, three Gs, and four Ts?

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