• (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 2,730 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 below:

• 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 your results.

• (HW_7_2) Tuberculosis (TB) is known to be a highly contagious disease. In 1995 a study was carried out on a random sample of 1074 Spanish prisoners. The study investigated factors that might be associated with the tuberculosis infection. The results follow.

Variable		PrisonersWithTuberculosis	Total Number Of Prisoners
Gender	Male	556	984
	Female	36	90
Race	White	496	886
	Gypsy	74	152
	Other	22	36
IntravenousDrugUsers	Yes	361	629
	No	231	445
HIV_Positive	Yes	186	294
	No	406	780
Re-imprisonment	Yes	272	456
	No	320	618


 
Is there any evidence to suggest that the race of the prisoner (White or Gypsy) makes any difference to whether they contracted tuberculosis? Carry out a significance test to answer this question and then calculate an appropriate 95% confidence interval.

Let p_W be the proportion of White prisoners infected with TB and p_G be the proportion of Gypsy prisoners infected with TB.

• Identify the parameter t (theta).
• State the hypotheses.
• Write down the estimate and its value.
• Find the P-value.
• Interpret the P-value.
• Answer the original question.
• Calculate a 95% confidence interval for the parameter.
• Interpret the 95% confidence interval.

• (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 500-
  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 answer.
  • 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.
  
         Age
  

  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
  Total	39,354	44,460	34,059	22,255	32,086	172,214

• 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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