Ivo Dinov
UCLA Statistics, Neurology, LONI
, UCLA Statistics
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STAT 13 (2a, 2b, 2c)

Introduction to Statistical Methods for the Life and Health Science

Department of Statistics

## Instructor: Ivo Dinov

Homework 6
Due Date: Friday, Nov. 21, 2003

Please, submit your homework right before lecture on the due date. See the HW submission rules. On the front page include the following header.

• (HW_6_1) Fluctuations in the market price of precious metals, such as gold, have been empirically shown to be well approximated by a Normal distribution when observed over short intervals of time. In a particular month, the daily price of gold (for 100g) was believed to have a mean of \$1250 and a standard deviation of \$28. A manufacturing jeweler will use 400g of gold during that particular month. She places the orders for the gold at the Monday prices each week. She wants to compare the differences in two different plans for buying the gold.
Plan 1:  Buy all 400g on the first Monday at the start of the month.
Plan 2:  Buy 100g of gold on each of the 4 Monday's of that month.

She has budgeted \$5100 to buy the gold. Let T1 be the total amount paid for the gold using plan 1 and T2 be the total amount paid for the gold using plan 2.

• Give an appropriate model (including parameters) for the distribution of T1 .
• Give an appropriate model (including parameters) for the distribution of T2 .
• How do the two models above differ?
• What is the probability of spending more than the budgeted amount using plan 1?
• What is the probability of spending more than the budgeted amount using plan 2?
• Which plan appears to be safer?

• (HW_6_2) Five fair, but unusual, octahedral (8 face) dice are rolled twice. Let Y be the random variable representing the total sum from this experiment (each die turns up an integer value between 1 and 8). Find the expected value of Y, mY, and the standard deviation of Y, sY.  Suppose, we carry this experiment (rolling five dice twice) 9 times. What would be approximately the distribution of the sample average, Y¯? What are the mean and the standard deviation of the sample average? [Note: Not all types of dice, any number of faces, are possible. For a nice description of which ones that are possible see the following web-page.]

• (HW_6_3) One of the oldest mnemonic techniques is known as the method of loci. Past experience suggests this method could be a useful technique for students who want to be able to remember an ordered set of points for an essay type question in an exam. The first step in this technique is to think of a set of locations that are naturally ordered. For example the route you commonly use to get from home to UCLA. Along this route are various familiar locations such as the bus stop where you catch the bus. You will require as many familiar locations as the number of points you wish to remember. The next step is to associate (in order) each location with one item you wish to remember. When you need to recall the points simply travel (in your mind) along the route, retrieving each item as you reach its location.
An experiment with two methods of instruction (normal and mnemonic) was conducted to evaluate the mnemonic method. Thirty-nine participants were randomly assigned to one of the two methods. Nineteen were given 24 words and instructed simply to "try and remember" each so that they could later recall the words (Normal controls). The second group of 20 participants studied the same 24 words but were instructed in the method of loci (Mnemonic group). One day later, participants were tested by being asked to recall the list of 24 words. The response measured was the number of words correctly recalled out of 24. The results were as follows:

 Controls: 9 7 5 9 7 5 11 7 9 10 15 13 13 8 7 12 12 7 17 Mnemonic group: 11 14 14 16 12 16 9 13 14 15 17 15 14 20 14 15 14 16 13 10

• What are the mean and standard deviation for the group of participants who received mnemonic instructions? How about the mean and standard deviation for the group of participants who received normal instructions?

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• Find a 95% confidence interval for the difference in the mean number of words recalled between the normal and mnemonic instruction methods. Interpret your result using plain English in 1-2 sentences.

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• What sample-size would yield a CI of half the size you found above?

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• Does your confidence interval above contain the true difference? Explain your answer briefly using plain English in 2-3 sentences.

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