Deciding when a distribution should be approximately normal By now, you should h
ID: 3362714 • Letter: D
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Deciding when a distribution should be approximately normal By now, you should have a good idea of when you should expect a distribution to be approximately normal as a result of the Central Limit Theorem. Recall the following rules of thumb that we have learned in class: The sum or average of n i.i.d. random variables should have approximately a normal distribution if n > 30 binomial distribution can be well approximated by a normal distribution if np2 10 A and n(l-p) 2 10. The same rule applies to a sample proportion. Recall also that individual observations could have many different distributions (such as binomial, geometric, Poisson, exponential, uniform, or normal). If we make a histogram of data that come from, say, an exponential distribution, then the histogram will show an exponential shape, not a normal distribution, regardless of how many observations there are Only averages (and therefore sums) of large numbers of observations are guaranteed to have approximately a normal distribution by the Central Limit Theorem. For each of the questions below, a histogram is described. Indicate in each case whether you think the histogram should look like approximately a bell-shaped (normal) curve, and give a brief explanation why (one sentence is probably sufficient). There are no data for these questions, so you will not need to use the computer to answer these q 13. A police department records the number of 911 calls made each day of the year, and the 365 values are plotted in a histogram. 14. The day before an election, fifty different polling organizations each sample 500 people and record the percentage who say they will vote for the Democratic candidate. The 50 values are plotted in a histogram. 15. The fifty polling organ izations also record the average age of the 500 people in their sample, and the 50 averages are plotted in a histogram. 16. One hundred batteries are tested, and the lifetimes of the batteries are plotted in a histogram. 17. Two hundred students in a statistics class each flip a coin 50 times and record the number of heads. The numbers of heads are plotted in a histogram. 18. Two hundred students in a statistics each roll a die 40 times and record the sum of the numbers they got on the 40 rolls. They make a histogram of the 200 sums. 19. One thousand randomly chosen people report their annual salaries, and these salaries are plotted in a histogramExplanation / Answer
13. There are 365 data points for phone calls per day. Since the number of observations > 30 , hence by CLT, the histogram should look like a normal curve.
14. There are 50 data points each of which is sample proportion of number of people voting for Democratic group. But each of the observation is a sample proportion and we have no idea about the value of n*p or n*(1-p). Hence we cannot comment if the curve will look like an histogram.
15. The average age in each of the 50 groups is plotted. Age is a continuous variable and there are more than 30 data points. So the histogram will look like a normal plot.
16. The number of data points are 100 >>30. and the variable observed is a continuous variable. (not a proprtion) and hence by CLT the curve should resemble a Normal probability plot.
17. The number of heads is dependant on the probability of heads of that coin. and unless we know 'p', we do not know if n*p > 10 or not. So in general the curve might not look like a normal curve.
18. Again like in (17), the sum of faces is dependant on the prob of each face and unless we know the different probabilities, we do not know the exact distribution of the sum of faces and hence we cannot comment on the curve.
19. The number of data points is 1000 and there is no probability distribution of Binomial type attached, so when plotted it will resemble a normnal curve.
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