Practice Test, Section 3 Spring 2017

Instructions: unless otherwise stated in the question, all answers of probability should be stated in percentage form to one decimal place. If the probability starts with zeros, make sure to have at least one non-zero digit. Unless the question specifically states otherwise, you can use any mathematical technique to solve the problem.

Probability Rules:

P(A or B) = P(A) + P(B) – P(A and B)

P(Ac) = 1 – P(A)

P(A and B) = P(A)P(B|A) = P(B)P(A|B)

P(B|A) = P(A and B) / P(A)

Discrete Random Variables: Binomial Random Variable:

Poisson Random Variable: z-scores

Confidence Intervals can be reported as LB < μ < UB or ( LB , UB )

For Linear Regressions, the degrees of freedom equals n-2

Regression:

Determining n for sample mean confidence interval, standard deviation known:

Constructing a Confidence Interval

Point Estimate Margin of Error

Determining n for sample population proportion:

(note: if an estimate of is not available, then use a default = .5)

  1. Regression

Year / Per capita consumption of margarine / Divorce rate in Maine / Predicted Divorce rate / Residual
2000 / 8.2 / 5
2001 / 7 / 4.7
2002 / 6.5 / 4.6
2003 / 5.3 / 4.4
2004 / 5.2 / 4.3
2005 / 4 / 4.1
2006 / 4.6 / 4.2
2007 / 4.5 / 4.2
2008 / 4.2 / 4.2
2009 / 3.7 / 3.7

Regress the Divorce rate in Maine (Y) on Per capita consumption of margarine (X) (data is real)

a)  Calculate the predicted Y for all the observations and enter in the chart above.

b)  Calculate the residual for all the observations and enter in the chart above.

c)  Write out the regression line:

d)  Find and interpret the r & r2:

e)  Find and interpret b:

f)  Find and interpret the Y intercept:

g)  Find and interpret the X intercept:

h)  If in 2015, people ate 2.6 pounds of margarine, what is the predicted divorce rate?

Conduct a hypothesis test to see if the consumption of margarine has any impact on the Divorce rate in Maine. Use α=.02 significance level. Note: degrees of freedom = n - 2

i)  Using the critical value method, what is/are the critical value(s), and which distribution is being used?

j)  Using the critical value method, what is the result (statistically) of the hypothesis test and why?

k)  Draw a diagram to represent the previous test

l)  Use the p-value method to conduct a hypothesis test

m)  State in English your results.

  1. A random sample of 300 college students was conducted. The students were asked if they had watched the TV show “Angry Housewives of Aptos”. 113 of them said yes, they had.
  1. What is the point estimate for the population proportion of college students who have seen the show?
  2. Construct a 95% confidence interval for the population proportion
  3. What is the margin of error?
  4. How many students would need to be sampled to have a 1% margin of error while maintaining the 95% confidence interval?
  1. (10 points) Central Limit Theorem
  1. What does the Central Limit Theorem say, and why is it so important?
  2. The height of a maple tree is distributed normally with a mean of 31 meters and a standard deviation of 4 meters. What is the probability of a tree being taller than 33 meters? Represent this graphically
  3. A group of 20 trees is selected at random. What is the probability that the average height of these 20 trees is more than 33 meters? Represent this graphically.
  4. A group of 20 trees is selected at random. What is the probability that the average height of these trees is between 30 and 32 meters? Represent this graphically.
  5. Calculate the Standard Error of the Mean
  1. A random sample of 65 households is conducted, and they are asked about how much they spend on vacations and travel. The sample mean is $1,780. The population has a standard deviation of σ = $450.
  2. Construct a 95% confidence interval for the population mean.
  3. What assumption is made to create this confidence interval?
  4. The population data is known to be heavily skewed to the right (the very rich spend a lot). Does this invalidate your results?
  5. What is the margin of error?
  6. How many household would need to be surveyed for the margin of error to be $50 (at the same confidence level)?
  7. A sample of beers that were bought at a particular bar were measured for their volume. The beers should be 16 ounces. Use Data Set A to represent the sample.

Data Set A: 15.7, 15.8, 16.0, 15.5, 15.7, 15.9, 16.3, 15.1, 15.4, 15.9, 16.0, 15.9, 16.1, 15.8, 15.5, 15.4, 15.8, 15.7, 16.2, 15.6

  1. Construct a 98% confidence interval for the population mean.
  2. What assumption is made to create this confidence interval?
  3. What is the margin of error?
  4. What does it mean that you are 98% confident in that interval?
  1. Use Data Set A above. A sample of beers that were bought at a particular bar were measured for their volume. Test whether the average beer was less than 16 ounces. Use α=.01 significance level.
  2. State the null and alternative hypothesis
  3. Using the critical value method, what is/are the critical value(s), and which distribution is being used?
  4. Using the critical value method, what is the result (statistically) of the hypothesis test and why?
  5. Draw a diagram to represent the previous test.
  6. Using the P-value method, what is the result (statistically) of the hypothesis test and why?
  7. State, in English, the result of the hypothesis test
  8. 150 randomly selected voters were surveyed. 81 of the voters said they would vote “yes” on Proposition O and/or P. Use α=.01 significance level. Conduct a hypothesis test to see if a majority of voters will pass the propositions.
  9. What conditions must hold make valid conclusions and are these conditions met?
  10. State the null and alternative hypothesis
  11. Using the critical value method, what is/are the critical value(s), and which distribution is being used?
  12. Using the critical value method, what is the result (statistically) of the hypothesis test and why?
  13. Draw a diagram to represent the previous test.
  14. Using the P-value method, what is the result (statistically) of the hypothesis test and why?
  15. State, in English, the result of the hypothesis test
  16. Use Data Set A above. A sample of beers that were bought at a particular bar were measured for their volume. Test whether the standard deviation of the pours is greater than .2 ounces. Use α=.01
  17. State the null and alternative hypothesis.
  18. Using the critical value method, what is/are the critical value(s), and which distribution is being used?
  19. Using the critical value method, what is the result (statistically) of the hypothesis test and why?
  20. Draw a diagram to represent the previous test.
  21. State, in English, the result of the hypothesis test
  22. What kind errors can be made when doing hypothesis testing, and how do we control those errors?
  23. What are the different probability distributions used in hypothesis testing and under what conditions are each used?