Normal Probabilities

Practice

The R function pnorm() will be very helpful to use for calculating probabilities. In order to use it we need to tell the function a few specific things we are using:

  • The value we are checking a probability for
  • The mean of the normal distribution
  • The standard deviation of the normal distribution
  • Whether to give the probability less than or more than (by default R returns ‘less than’ probabilities, use “lower.tail=F” to get ’greater than probabilities)

Here is an example of using the pnorm() function to calculate the probability P(X > 25) for a normal distribution with mean 30 and std.dev. 10.

pnorm(25, mean=30, sd=10, lower.tail = FALSE)
## [1] 0.6914625
# Nice

Question 1

We will use a standard normal distribution for practice. For all of these question parts, show your R code and your calculations.

Part A: Use R to find the probability of randomly picking a value more than 0.5.

Part B: Use R to find the probability of randomly picking a value less than -0.5.

Part C: Do your answers to Parts A and B match?

Part D: Use Parts A and B to find the probability of randomly picking a value between -0.5 and 0.5.

Part E: Verify the 68-95-99.7% rule, and state the results in your own words.

Practical Example – Mercury in Fish

Mercury is a chemical that is toxic to humans (and many others animals). From the smokestacks of power plants to the discharges from wastewater treatment plants, and other places, mercury in the form of the compound methylmercury exists in the enviroment, and it can settle to the seafloor and be taken up by tiny organisms that live or feed on bottom sediments.

These compounds aren’t digested, they accumulate within the animals that ingest them, and become more and more concentrated as they pass along the food chain as animals eat and then are eaten in turn. This is biomagnification, and it means that higher-level predators-fish, birds, and marine mammals-build up greater and more dangerous amounts of toxic materials than animals lower on the food chain. (Info taken from here)

The U.S. Food and Drug Administration recommends avoiding eating fish with mercury levels higher than 0.46 \(\mu\)g/g (micro-grams mercury per gram of fish), as they may be harmful (source).

Suppose the population of yellowfin tuna follows a Normal distribution with an average mercury level of 0.354 \(\mu\)g/g , and a variance of 0.02. (Mean value taken from here, I made up the variance amount – hard to find this)

Question 2:

Part A: What is the standard deviation of this population?

Part B: What is the probability of catching a yellowfin tuna with an unsafe amount of mercury? Would you frequently eat yellowfin tuna knowing this?

Part C: Suppose the standard deviation is acutally .03. Find the probability of catching a yellowfin tuna with an unsafe amount of mercury. Does this change your answer to whether or not you would frequently eat yellowfin tuna?

Useful Website Stuff

Sampling Distribution

Go to [this website] and go through the tutorial and associated examples. It will help explain what the sampling distribution is all about.

Question 3: What are some things you learned from this example?

Central Limit Theorem

Go to [this website] and go through the tutorial and associate examples. It will help explain what the CLT is all about.

Question 4: What are some things you learned from this example?

Confidence Intervals

Florida Mercury Levels

This is using the info from a study similar to the Florida lakes fish data we saw in the slides.

Data was collected from 107 lakes in Florida. For each lake, the mercury level (ppm) was computed for a large mouth bass. The average mercury level in the sample was 0.554 ppm.

Question 5

Part A: Describe the parameter in context. If we know the value, provide it.

Part B: Describe the statistic in context. If we know the value, provide it.

Part C: The standard error for this statistic is 0.051. What is the 95% confidence interval? (Show your work)

Part D: Interpret the confidence interval.

Part E: In the U.S., the FDA action level is 1 ppm. Is this safely below the U.S. limit?

Part F: In Canada, the safety limit is 0.5 ppm. Is this clearly above the Canadian limit?

Part G: Suppose we didn’t know the standard error value in part C, but instead we knew the population standard deviation is 0.20. Create a new 95% confidence interval using this info. Does your answer to Part F change?

Question 6 (Conceptual)

Part A: What is the purpose of a confidence interval?

Part B: What does “95% confidence” actually mean?

Part C: Do confidence intervals account for biased samples?