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# Statistics: Standard Deviation, Random Chance and Unusual Results

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[2] Neuroblastoma, a rare form of malignant tumor, occurs in 11 children in a million, so its probability is 0.000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12,429 children.
a. Assuming that neuroblastoma occurs as usual, find the mean and standard deviation of cases in groups of 12,429 children.
b. Find the probability that the number of neuroblastoma cases in a group of 12,429 children is 0 or 1.
c. Does the cluster of four cases in Oak Park, Illinois, appear to be attributable to random chance? Why or why not?

[4] A recent Gallup poll consisted of 1012 randomly selected adults who were asked whether "cloning of humans should or should not be allowed." Results showed that 89% of those surveyed indicated that cloning should not be allowed.
a. If we assume that people are indifferent so that 50% believe that cloning of humans should not be allowed, find the mean and standard deviation for the numbers of people in groups of 1012 that can be expected to believe that such cloning should not be allowed.
b. Based on the preceding results, does the 89% result for the Gallup poll appear to be unusually higher than the assumed rate of 50%? Does it appear that an overwhelming majority of adults believe that cloning of humans should not be allowed?

https://brainmass.com/statistics/quantative-analysis-of-data/statistics-standard-deviation-random-chance-unusual-results-503014

#### Solution Preview

[2] Neuroblastoma, a rare form of malignant tumor, occurs in 11 children in a million, so its probability is 0.000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12,429 children.

a. Assuming that neuroblastoma occurs as usual, find the mean and standard deviation of cases in groups of 12,429 children.

Let X be the cases of neuroblastoma in groups of 12,429 children. Then X follows a binomial distribution with n=12429 and p=0.000011. So,

Mean = E(X) = np = 12429*0.000011= 0.1367
Standard deviation = sigma(X) = root of [np(1-p)] = root of [12429*0.000011*0.999989] = 0.3698

b. Find the ...

#### Solution Summary

Standard deviation, random change and unusual results are examined.

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## Solving Multiple Questions on Descriptive Statistics

1. Determine whether the description corresponds to an observational study or an experiment.
Forty patients with skin cancer are divided into two groups. One group an experimental drug to fight cancer, the other a placebo. After two years, the spread of the cancer is measured.

Does the description correspond to an observational study or experiment?

2. A worker assembles one item every 20 minutes, so 120 items are completed in her first week of work. Her manager checks her work by randomly selecting a day of the week, then reviewing all the items she completed that day. Does the sampling plan result in a random sample? Simple random sample?

Does this sampling plan result in a random sample?
Does this sampling plan result in a simple random sample?

3. Find the mean median mode and midrange for the given sample data in millions of dollars. Given that these the top 10 salaries, do we know anything about the salaries of TV personalities in general? Are such top 10 lists valuable for gaining insight into the larger population?
37.4 36 35.6 26.5 15.3 12.5 12.1 9.2 9.7 8
A. Mean___
B. Median____
C. Mode _____
D. Midrange_____

Given that these are the top 10 salaries, do we know anything about the salaries of TV personalities in general?

Are such top 10 lists valuable for gaining insight into the larger population?

4. A woman wrote to a newspaper advice columnist and claimed that she gave birth 313 days after a visit from her husband, who was in the Navy. Lengths of pregnancies have a mean of 267.6 days and a standard deviation of 15.5 days. Find the z score for 313 days. Is such a length unusual?
The z score is ___
Is the pregnancy length of 313 days unusual?

5. Suppose a baseball player had 208 hits in a season. In the given probability distribution, the random variable X represents the number of hits the player obtained in a game.
x 0 1 2 3 4 5
P(x) 0.1595 0.4758 0.2258 0.0872 0.0449 0.0068
a) Compute and interpret the mean of the random variable X.
µx=____
What is a good interpretation of the mean?
b) Compute the standard deviation of the random variable X.
Ơx=____.

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