Introduction to Statistical Methods for the Life and Health Sciences
Instructor: Ivo Dinov, Asst. Prof.
Departments of Statistics & Neurology
Friday, Mar. 14, 2003, turn in after lecture
HW submission rules.
On the front page include the following
(HW_7_1) In 1996 the New Zealand Consumers' Institute conducted
a survey on home computer use. 7,400 subscribers to Consumer Magazine
were randomly selected and sent a survey form. Of those surveyed
had a computer for personal use at home. The respondents who had a home
computer were given a list of computer activities and were asked to indicate
all of those that they engaged in. They were also asked to indicate the
number of hours per week that they used their computer. The Consumers'
Institute used the results to draw conclusions about subscribers who
own a home computer. The results of the survey are given in the two tables
||Computer Use (Hours Per Week)
||Used For Less Than 2 Hours
||Over 2 and Up To 7 hours (incl)
||Over 7 and Up To 14 hours (incl)
||Over 14 and Up To 21 hours (incl)
||Over 21 and Up To 28 hours (incl)
||Over 28 and Up To 35 hours (incl)
||Over 35 and Up To 44 hours (incl)
||Over 44 hours
Is this survey representative of the home computer use of New Zealanders?
Briefly justify your answer.
What proportion of the respondents use their computers for drawing?
What percentage of the respondents use their computers between 21 and 28
hours (inclusive) per week?
Is there any evidence to suggest from the survey that there is a significant
difference between the proportion of respondents who use their computer
for drawing and the proportion of respondents who use their computer for
desktop publishing. Carry out a Z-test to investigate this. Calculate a
95% confidence interval for the difference in these proportions. Interpret
- (HW_7_3) The manager of an importing company purchased a new
machine for packaging rice. The specifications of the machine claim that
the amount of rice put in each package will be Normally distributed with
an average amount of rice as specified and a standard deviation of 2.7 grams.
The machine is set to fill packets with 506 grams of rice. The manager requires
the machine to produce packets containing rice weighing within the range
512 grams for at least 95% of the packets.
- Assuming that the specifications for the machine are accurate,
calculate the probability that a packet of rice contains the desired
amount of rice. Will the managers requirements be met?
- Let X be the mean weight from a sample of 25 packets of rice. Assume that the specifications for the machine are accurate.
- Give the distribution and parameters for X .
- Was the central limit theorem needed to answer the question above? Briefly justify your answer.
- Calculate the interval within which the central 95% of values of X should fall.
- Let Pˆ be the proportion of packets of rice outside the 500 –
512 gram weight range from a sample of 800 packets of rice. Assume that
the specifications for the machine are accurate.
- Give the distribution and parameters for Pˆ.
- Was the central limit theorem needed to answer the question above?
- Calculate the interval within which the central 95% of values of Pˆ should fall.
- A sample of 25 packets of rice was filled and weighed. Assume
the sample size = 25, sample mean = 504.5 grams, sample standard deviation
= 3.93 grams. The resulting weights were as follows:
|502.8|| 507.6|| 515.0|| 499.2|| 511.9|| 503.4|| 506.3|| 505.5|| 502.6|| 501.9|
|500.8|| 502.9|| 503.9|| 510.3|| 502.4|| 502.9|| 505.4|| 510.4|| 501.6|| 501.4|
|498.3|| 504.0|| 504.4|| 504.5|| 503.9||
- Create a stem-and-leaf plot of the data (either by hand or by computer).
- Does the sample of weights appear to have an exactly Normal distribution? Briefly justify your answer.
- Are there any features of the stem-and-leaf plot that suggest
major departures from the Normal distribution? Briefly justify your
- Based on the above answers, is it likely that the sample data has come from a Normal distribution?
- Assuming that the specifications of the machine are correct, would
it be unusual to get a sample of 25 packets of rice with a mean weight
of 504.5 grams Does getting a sample of 25 packets of rice with a mean weight
of 504.5 grams cast any doubts on the specifications of the machine? [Hint:
refer to part (b).]
- A sample of 800 packets of rice are produced. Of these, 26 were found to be outside the 500 – 512 gram weight range.
Assuming that the specifications of the machine are correct,
would it be unusual to get a sample of 800 packets of rice with 26 of the
packets falling outside the 500 – 512 gram weight range. Does getting a sample
of 800 packets of rice with 26 of the packets falling outside the 500 – 512
gram weight range cast any doubts on the specifications of the machine? [Hint:
refer to part (c).]
- Write a brief report (a couple of paragraphs) discussing whether
or not the given specifications of the machine appear to correct.
- (HW_7_4) The U.S. Bureau of the Census recently published statistics
on educational attainment of the non-institutional population of the United
States, based on the March 1998 Current Population Survey. 172,214 people
were surveyed and classified by age group and highest educational qualification
attained. The following table summarises the results of the survey.
|Level of Education||25–34||35–44||45–54||55–64||Over-64||Total|
|Did not complete high school||4,754||5,326||4,341||4,558||10,580||29,559|
|Completed high school||12,569||15,136||10,943||8,311||11,215||58,174|
|Attended university for between 1 and 3 years||19,587||20,450||14,921||7,379||8,478||70,815|
|Attended university for 4 or more years||2,444||3,548||3,854||2,007||1,813||13,666|
- Which age category accounted for the:
- lowest number of Americans who did not complete high school?
- the highest percentage of Americans who attended university for at least one year. Show your work.
- What percentage of Americans in this survey:
- did not attend university?
- who only completed high school or were aged between 35 and 44 years?
- Given that a randomly chosen American in this survey had only
completed high school, what is the probability that he or she is at most
54 years of age?
- Among Americans in this survey aged between 25 and 34 years what proportion did not attend university?
- Is the proportion of people ages 25-34 who attended university
for 4 or more years statistically different form the proportion of people
over-64 that did not complete high-school?
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Ivo D. Dinov, Ph.D., Departments of Statistics and Neurology,
UCLA School of Medicine