# Case Study For Normal Distribution

Subject: StatisticsBook: Complete Business Statistics, Aczel & Sounderpandian ( sixth ed)Chapter 4: The Normal DistributionProblem 4-61Scores on a management aptitude examination are believed to be normally distributed with mean 650(out of a total of 800 possible points) and standard deviation 50. What is the probability that arandomly chosen manager will achieve a score above 700? What is the probability that the score will bebelow 750?Problem 4-72Total annual textbook sales in a certain discipline are normally distributed. Forty-five percent of thetime, sales are above 671,000 copies, and 10% of the time, sales are above 712,000 copies. Find themean and the variance of annual sales.Chapter 5: Sampling and Sampling DistributionsProblem 5-56Explain why the sample variance is defined as the sum of squared deviations from the sample mean,divided by

n

- 1 and not by

n.

Problem 5-71A new kind of alkaline battery is believed to last an average of 25 hours of continuous use(in a given kind of flashlight). Assume that the population standard deviation is 2 hours. If a randomsample of 100 batteries is selected and tested, is it likely that the average battery in the sample will lastless than 24 hours of continuous use?Chapter 6: Confidence IntervalsProblem 6-71The legal blood alcohol limit in France is 0.05 percent. Recently, the French have begun doingsomething they never would have considered before: taking home half-empty bottles of wine in orderto avoid exceeding the blood alcohol limit when driving home from a restaurant. If the following are theblood alcohol levels of a random sample of French diners who take home some of their wine, constructa 95% confidence interval for the blood alcohol level of the population of such people.0.01, 0.07, 0.09, 0.03, 0.03, 0.005, 0.008, 0.01, 0.08, 0.04, 0.03, 0.02, 0.04Based on your answer, does the average French diner who takes home a bottle of wine break the law?Problem 6-89

Let's imagine, we start with one single bacterium. At each time step (generation), each bacterium has $x$ offspring and it dies (semelparous species). $x$ is a value drawn from a normal distribution with mean=$M$, standard deviation=$SD$.

**Question 1:**

What is the probability distribution of the number of bacteria (population size) after t generations ?

**Question 2:**

Same question but assuming that nobody ever dies ! So that after the very first reproductive event, there are $n$ bacteria, value which is drawn from a normal distribution mean = $M+1$, standard deviation = $SD$.

FastingGuy says the the probability distribution is normal with mean = $M*t$, standard deviation = $sqrt(t)*SD$.

Below is a very simple R-script that shows that the population size is almost always lower when $SD$ is higher and therefore the mean should depend of $SD$. What am I missunderstanding. In my code I'm using a uniform distribution to avoid a crash because all individuals reproduce equally at each generation. Is this the reason that higher $SD$ yields to lower population size ?

**Y is in logarithmic scale !**

Thank you.

statisticsprobability-distributionsnormal-distribution

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