12A Preview

Preparing for the next class 

In the next in-class activity, you will need to use the DCMP Normal Distribution tool to  calculate normal probabilities, simulate random samples from a population using the  DCMP Sampling Distribution of the Sample Mean (Continuous Population) tool, find the  mean and standard deviation of the sampling distribution of the sample mean, and use  the mean and standard deviation of the sampling distribution of the sample mean to  calculate and interpret a z-score for a sample mean.

The SAT is an assessment designed to evaluate a student’s college-specific skills. SAT  scores tend to follow an approximate normal distribution with mean of 1060 and a standard deviation of 195.[1]

Go to the DCMP Normal Distribution tool at https://dcmathpathways.shinyapps.io/NormalDist/.

  • Select the Find Probability tab.
  • Enter 1060 for the mean, [latex]\mu[/latex].
  • Enter 195 for the standard deviation, [latex]\sigma[/latex].

The normal distribution displayed now represents a model for the distribution of the SAT  scores for all students who take the exam.

Question 1

1) Use the tool to find the following probabilities.

a) What is the probability that a randomly selected student scores higher than a 1200 on the SAT?

b) What proportion of students scores between 865 and 1255 on the SAT? Hint: Your answer should be consistent with the Empirical Rule.

Question 2

2) Instead of looking at the distribution of the SAT scores for individual students, you  are interested in the distribution of the average SAT scores across classrooms.  Suppose that the class size at a large high school is 10 students per class.

a) If you were to calculate the probability that the average SAT score for a  randomly selected classroom is larger than 1200, would you predict this  probability to be smaller, larger, or approximately the same as your answer  to Question 1, Part A?

a) Smaller

b) Larger

c) Approximately the same

b) If you were to calculate the proportion of classrooms with average SAT  scores between 865 and 1255, would you predict this probability to be  smaller, larger, or approximately the same as your answer to Question 1, Part B?

a) Smaller

b) Larger

c) Approximately the same

Let’s use simulation to check your predictions from Question 2. Go to the DCMP  Sampling Distribution of the Sample Mean (Continuous Population) tool at  https://dcmathpathways.shinyapps.io/SampDist_cont/.

  • Under “Select Population Distribution,” choose “Bell-Shaped.”
  • Set the population mean to 1060 and the population standard deviation to 195.  (You will need to select the “Enter values for µ and σ” option.)

The population distribution displayed now represents a model for the distribution of the individual SAT scores for all students who take the exam.

Question 3

3) Generate 1,000 samples of size 10 from this population. The simulated distribution shown at the bottom is the sampling distribution of the sample mean—in this case, the distribution of the average SAT scores for your 1,000 randomly generated  classrooms of 10 students. Select the “Find Probability” box and answer the following questions.

a) What proportion of your simulated classrooms have an average SAT score greater than 1200? Does this match your prediction from Question 2, Part A?

Hint: The data analysis tool will report the proportion of simulations resulting in a sample mean less than or equal to 1200. You will need to subtract that proportion  from 1 to get your answer.

b) What proportion of your simulated classrooms have an average SAT score between 865 and 1255? Does this match your prediction from Question 2, Part B?

Hint: You will need to find the proportion of simulations at or below 865 and then subtract that value from the proportion of simulations at or below 1255.

c) Select the “Show Normal Approximation” box. This overlays a normal distribution with the same mean and standard deviation as the distribution of the simulated average SAT scores. What are the mean and standard deviation of your simulated average SAT scores? Does the distribution of the simulated average SAT scores seem to follow the normal curve?

When sampling from a normal population such as SAT scores, the distribution of the sample means will also have a normal distribution with the same mean; but, the variability in sample means will be less than the variability in individuals (similar to how  variability in sample proportions will be less than the variability in individuals). There are  mathematical formulas we can use to find the mean and standard deviation of the  sampling distribution of the sample mean for samples of size [latex]n[/latex]:

Mean of the sampling distribution of the sample mean [latex]=\mu[/latex]

Standard deviation of the sampling distribution of the sample mean [latex]\frac{\sigma}{\sqrt{n}}[/latex]

In the formulas above, [latex]\mu[/latex] and [latex]\sigma[/latex] represent the mean and standard deviation of the original population, respectively. For the population distribution of the individual SAT scores, [latex]\mu = 1060[/latex] and [latex]\sigma = 195[/latex].

Question 4

4) Use the formulas above to calculate the mean and standard deviation of the sample  mean SAT scores for samples of size [latex]n=10[/latex]. Round to the nearest thousandth.

Hint: These values should be close to the simulated mean and standard deviation you found in Question 3, Part C.

Question 5

5) Go back to the DCMP Normal Distribution tool at https://dcmathpathways.shinyapps.io/NormalDist/.

  • Select the Find Probability tab.
  • Enter the mean you found in Question 4.
  • Enter the standard deviation you found in Question 4.

The normal distribution displayed now represents a model for the distribution of the mean SAT scores for all classrooms of 10 students who take the exam. Use the tool to find the following probabilities.

Part A: What is the probability that the mean SAT score for a randomly selected  classroom of 10 students is higher than 1200? Is this value similar to your  answer to Question 3, Part A?

Part B: What proportion of classrooms of 10 students have an average SAT score  between 865 and 1255? Is this value similar to your answer to Question 3, Part B?

Question 6

6) Suppose one classroom’s average SAT score is 950. Use the mean and standard  deviation you found in Question 4 to calculate the z-score for a sample mean SAT  score of 950. Write a sentence interpreting this value in context of the problem.

  1. Institute of Education Sciences, National Center for Education Statistics. (2019). Digest of education  statistics 2019 tables and figures: Table 226.40. https://nces.ed.gov/programs/digest/d19/tables/dt19_226.40.asp ↵