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Introductory Statistics, Math 203 B, Quiz 5, Tom Linton
Work alone, open book-notes, due 9 AM April 17, 2000

  1. A trout hatchery produces 2-year old trout that average 15.4 inches in length with a standard deviation of s = 2.3 inches (assume that the lengths of trout in this hatchery are normally distributed). The hatchery then decides to make two significant changes to their rearing procedures. They switch from a high quality fish food to a lower quality fish food (that may lower the growth rates) and install a fancy new water temperature control system (that should increase the growth rates). The hatchery wants to test whether there is any change in the lengths of their 2-year old trout due to these modifcations.
    1. Does this call for a right-tailed test, left-tailed test or two-tailed test?

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    2. Setup the appropriate null and alternative hypothesis.

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    3. After the modifications are made and the new rearing scheme is used for two years, 12 randomly selected 2-year old trout have an average length of  = 14.3 inches. Calculate the p-value of your test and discuss whether or not these modifications altered the average lengths of trout in this hatchery.

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  2. A significance test of H0: m = 16.5 against the alternative Ha: m < 16.5 yields a p-value of p = 0.027. To calculate this p-value a random sample of size n = 34 was used and the sample mean was  = 15.74.
    1. At the 5% level, do you reject H0 or accept H0?

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    2. Draw a rough sketch of a normal density curve, labeling all relevant values, that shows the area corresponding to this p-value.

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    3. Explain (carefully) in everyday language what this p-value tells you.

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  3. A trusted friend offers you an investment where you give them $1000 and at the end of each month, your friend pays you your earnings. After each 3-month period, you can continue the investment for another 3 months, or withdrawl your $1000. The monthly earnings are claimed to vary in a normal fashion with a mean return of $15.00 and a standard deviation of $4.00. Recognizing that your average annual return will be $180 (18% profit), you consider this a good deal and decide to invest $1000.
    1. Your first three returns, $12.05, $13.29 and $12.64 (so  = $12.66) are below average and you start to doubt the claim that the average monthly return is actually $15.00, thinking it is likely a smaller amount. However, the friend is a good one, so you'd like to be pretty sure that your 3-month average return wasn't just due to bad luck, before pulling your money out of the investment. This situation calls for a left-tailed significance test. Define the null and alternative hypothesis and calculate the p-value for this test.

      2.  

       
       
       
       
       
       
       
       
       
       
       

    2. The p-value from part (a) is large enough for you to trust your friend and leave your investment with them for 3 more months. You decide however that after 6 months (so n = 6 now), you will test the hypothesis that the average monthly return is $15.00 against the alternative that it is less than $15.00 at the 5% significance level. If the result is significant at the 5% level, you will withdrawl your investment (reject H0), otherwise you will leave your $1000 in for another 3-month period (accept H0). Calculate the acceptance interval, i.e. for which values of  (your average return after 6 months) will you leave your $1000 invested?

      3.  

       
       
       
       
       
       
       
       
       
       
       
       

    3. Which of the situations below corresponds to a type-2 error for the test in part (b)?
      1. Your friend was truthful about the average monthly return, but your test indicated that the true return was less than the quoted figure of $15.00.
      2. Your friend lied about the average return (it was actually less than $15.00), but the test in part (b) indicated that the quoted return of $15.00 a month was reasonable.
      3. There was a typo in the information about the average monthly return, and the 5 in $15.00 should have been a 2.

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    4. Calculate the probability of a type-2 error for the test in part (b) against the alternative that the average monthly return is actually $11.00.