Deciding when a distribution should be approximately normal By now, you should h
ID: 3363047 • Letter: D
Question
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,1230 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 questions. 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.
Since the number of calls would be hovering over a range. say 100 - 500 calls. Over a period of one year, you can expect the curve to be bell shaped, but you cannot conclusively say it.
It will not be bell shaped or will not be a normal distribution.
14.
the percentage who vote for democratic candidates will be usually near40-60% (same applicable for republic candidate). It will not be a wild swing for any of these candidates.
Even with a minimum p of 0.2 (20%) and n of 50 the np value is 10. so with p of approx above 40%, this binomial distribution can be approximated to normal distribution. It will be bell shaped.
15.
Since the distribution under consideration is 50 average scores (each average score is average of 500 people), the histogram will be normal distribution as per central limit theorem
16.
100 batteries are tested. No other information is given, it can be any form of distribution. Not necessarily normal.
17.
The coin toss is a binomial distribution.
When n = 200 and p = 0.5, np = 100 which is greater than 20. Hence this will be approximately normal distribution.
18.
The random variable is the sum of 40 rolls.
200 samples of the random variable (sum of 40 rolls) will be a normal distribution as per central limit theorem
19.
1000 report their annual salaries.
It will not be a normal distribution (not sure 100% it will be normal) because it is a not a distribution of sum or average or binomial distribution.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.