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Proportion and Quartiles

Please see the attached file for the questions.

1. A set of data whose histogram is bell shaped yields a sample mean and standard deviation of 50 and 4, respectively. Approximately what proportion of observations are:
a) Between 46 and 54?
b) Between 42 and 58?
c) Between 38 and 62?

2. Calculate the first, second, and third quartiles for the following sample.
5 8 2 9 5 3 7 4 2 7 4 10 4 3 5

3. There is a garbage crisis in North America - too much garbage and no place to put it. As a consequence, the idea of recycling has become quite popular. A waste-management company in a large city is willing to begin recycling newspapers, aluminum cans, and plastic containers. However, it is profitable to do so only if a sufficiently large proportion of households are willing to participate. In this city, 1 million households are potential recyclers. After some analysis it was determined that, for every 1,000 households that participate in the program, the contribution to profit is $500. It was also discovered that fixed costs are $55,000 per year. It is believed that 50,000, 100,000, 200,000, or 300,000 households will participate with probabilities of .5, .3, .1, and .1, respectively. A preliminary survey was performed where 25 households were asked whether they would be willing to be part of this recycling program.
Suppose only 3 of the 25 respond favorably. Incorporate this information into a decision-making process to decide whether the waste-management company should proceed with the recycling venture.

4. Suppose the following statistics were calculated from data gathered from a randomized block experiment with k = 4 and b = 10:
SS(total) = 1,210 SST = 275 SSB = 625
(a) Can we conclude from these statistics that the treatment means differ? (use α =.01)
(b) Can we conclude from these statistics that the block means differ? (Use α = .01)

5. A random sample of 50 observations yielded the following frequencies for the standardized intervals:
Interval Frequency

Z ≤ - 1 6

-1 < Z ≤ 0 27

0 < Z ≤ 1 14

Z >1 3
Can we infer that the data are not normal? Use α = .10.


Solution Summary

This solution is comprised of a detailed explanation for different questions on proportions, frequencies, chi-square test. All the formulas are given with respect to different questions such as mean, standard deviation, z score, probability calculation using standard normal distribution, quartile, median and chi-square test. Full explanantion is given for every question.