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8.15 Homework  (Page 24/8)

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This module provides a homework of Hypothesis Testing of Single Mean and Single Proportion as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Some of the statements below refer to the null hypothesis, some to the alternate hypothesis.

State the null hypothesis, H o size 12{H rSub { size 8{o} } } {} , and the alternative hypothesis, H a size 12{H rSub { size 8{a} } } {} , in terms of the appropriate parameter ( μ size 12{μ} {} or p size 12{p} {} ).

  • The mean number of years Americans work before retiring is 34.
  • At most 60% of Americans vote in presidential elections.
  • The mean starting salary for San Jose State University graduates is at least $100,000 per year.
  • 29% of high school seniors get drunk each month.
  • Fewer than 5% of adults ride the bus to work in Los Angeles.
  • The mean number of cars a person owns in her lifetime is not more than 10.
  • About half of Americans prefer to live away from cities, given the choice.
  • Europeans have a mean paid vacation each year of six weeks.
  • The chance of developing breast cancer is under 11% for women.
  • Private universities mean tuition cost is more than $20,000 per year.
  • H o : μ = 34 size 12{H rSub { size 8{o} } :μ="34"} {} ; H a : μ ≠ 34 size 12{H rSub { size 8{a} } :μ<>"34"} {}
  • H o : μ ≥ 100 , 000 size 12{H rSub { size 8{o} } :μ>= "100","000"} {} ; H a : μ < 100 , 000 size 12{H rSub { size 8{a} } :μ<"100","000"} {}
  • H o : p = 0 . 29 size 12{H rSub { size 8{o} } :p=0 "." "29"} {} ; H a : p ≠ 0 . 29 size 12{H rSub { size 8{a} } :p<>0 "." "29"} {}
  • H o : p = 0 . 50 size 12{H rSub { size 8{o} } :p=0 "." "50"} {} ; H a : p ≠ 0 . 50 size 12{H rSub { size 8{a} } :p<>0 "." "50"} {}
  • H o : p ≥ 0 . 11 size 12{H rSub { size 8{o} } :p>= 0 "." "11"} {} ; H a : p < 0 . 11 size 12{H rSub { size 8{a} } :p<0 "." "11"} {}

For (a) - (j) above, state the Type I and Type II errors in complete sentences.

  • Type I error: We conclude that the mean is not 34 years, when it really is 34 years. Type II error: We do not conclude that the mean is not 34 years, when it is not really 34 years.
  • Type I error: We conclude that the mean is less than $100,000, when it really is at least $100,000. Type II error: We do not conclude that the mean is less than $100,000, when it is really less than $100,000.
  • Type I error: We conclude that the proportion of h.s. seniors who get drunk each month is not 29%, when it really is 29%. Type II error: We do not conclude that the proportion of h.s. seniors that get drunk each month is not 29%, when it is really not 29%.
  • Type I error: We conclude that the proportion is less than 11%, when it is really at least 11%. Type II error: We do not conclude that the proportion is less than 11%, when it really is less than 11%.

For (a) - (j) above, in complete sentences:

  • State a consequence of committing a Type I error.
  • State a consequence of committing a Type II error.
For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in 14. Appendix (online book version: the link is "Solution Sheets"; PDF book version: look under 14.5 Solution Sheets). Please feel free to make copies of the solution sheets. For the online version of the book, it is suggested that you copy the .doc or the .pdf files.

If you are using a student's-t distribution for a homework problem below, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9800 miles. Do the data support the claim at the 5% level?
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Source:  OpenStax, Collaborative statistics (custom lecture version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11543/1.1
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