Data Science: Inference and Modelling part I (EXERCISE)

An Exercises from HarvardX: PH125.4x

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Exercise 1. Polling - expected value of S

Suppose you poll a population in which a proportion p of voters are Democrats and 1-p are Republicans. Your sample size is N=25. Consider the random variable S, which is the total number of Democrats in your sample.

What is the expected value of this random variable S?

Answer:

\(\mbox{E}(S) = 25p\)

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Exercise 2. Polling - standard error of S

Again, consider the random variable S, which is the total number of Democrats in your sample of 25 voters. The variable p describes the proportion of Democrats in the sample, whereas 1-p describes the proportion of Republicans.

What is the standard error of S?

Answer:

\(\mbox{SE}(S) = \sqrt{25 p (1-p)}\)

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Exercise 3. Polling - expected value of \(\bar{X}\)

Consider the random variable S/N, which is equivalent to the sample average that we have been denoting as \(\bar{X}\). The variable N represents the sample size and p is the proportion of Democrats in the population.

What is the expected value of \(\bar{X}\)?

Answer:

\(\mbox{E}(\bar{X}) = p\)

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Exercise 4. Polling - standard error of \(\bar{X}\)

What is the standard error of the sample average, \(\bar{X}\)?

The variable N represents the sample size and p is the proportion of Democrats in the population.

Answer:

\(\mbox{SE}(\bar{X}) = \sqrt{p (1-p) / N}\)

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Exercise 5. se versus p

Write a line of code that calculates the standard error se of a sample average when you poll 25 people in the population. Generate a sequence of 100 proportions of Democrats p that vary from 0 (no Democrats) to 1 (all Democrats).

Plot se versus p for the 100 different proportions.

Instructions

• Use the seq function to generate a vector of 100 values of p that range from 0 to 1.
• Use the sqrt function to generate a vector of standard errors for all values of p.
• Use the plot function to generate a plot with p on the x-axis and se on the y-axis.

Answer:

rr # N represents the number of people polled N <- 25 # Create a variable p that contains 100 proportions ranging from 0 to 1 using the seq function p <- seq(0, 1, length.out=100) # Create a variable se that contains the standard error of each sample average se <- sqrt(p * (1-p) / N) # Plot p on the x-axis and se on the y-axis plot(p, se)

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Exercise 6. Multiple plots of se versus p

Using the same code as in the previous exercise, create a for-loop that generates three plots of p versus se when the sample sizes equal N=25, N=100, and N=1000.

Instructions

• Your for-loop should contain two lines of code to be repeated for three different values of N.
• The first line within the for-loop should use the sqrt function to generate a vector of standard errors se for all values of p.
• The second line within the for-loop should use the plot function to generate a plot with p on the x-axis and se on the y-axis.
• Use the ylim argument to keep the y-axis limits constant across all three plots. The lower limit should be equal to 0 and the upper limit should equal the highest calculated standard error across all values of p and N.

Answer:

rr # The vector p contains 100 proportions of Democrats ranging from 0 to 1 using the seq function p <- seq(0, 1, length = 100) # The vector sample_sizes contains the three sample sizes sample_sizes <- c(25, 100, 1000) # Write a for-loop that calculates the standard error se for every value of p for each of the three samples sizes N in the vector sample_sizes. Plot the three graphs, using the ylim argument to standardize the y-axis across all three plots. for (val in sample_sizes) { se <- sqrt(p * (1-p) / val) plot(p, se, ylim=c(0,max(se))) }

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Exercise 7. Expected value of d

Our estimate for the difference in proportions of Democrats and Republicans is \(d = \bar{X} - (1-\bar{X})\).

Which derivation correctly uses the rules we learned about sums of random variables and scaled random variables to derive the expected value of d?

Answer:

\(\begin{eqnarray} \mbox{E}[\bar{X} - (1-\bar{X})] &=& \mbox{E}[2\bar{X} - 1] \ &=& 2\mbox{E}[\bar{X}] - 1 \ &=& 2p - 1\ &=& p - (1-p) \end{eqnarray}\)

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Exercise 8. Standard error of d

Our estimate for the difference in proportions of Democrats and Republicans is \(d = \bar{X} - (1-\bar{X})\).

Which derivation correctly uses the rules we learned about sums of random variables and scaled random variables to derive the standard error of d?

Answer:

\(\begin{eqnarray} \mbox{SE}[\bar{X} - (1-\bar{X})] &=& \mbox{SE}[2\bar{X} - 1] \ &=& 2\mbox{SE}[\bar{X}] \ &=& 2\sqrt{p(1-p)/N} \end{eqnarray}\)

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Exercise 9. Standard error of the spread

Say the actual proportion of Democratic voters is \(p=0.45\). In this case, the Republican party is winning by a relatively large margin of \(d= -0.1\), or a 10% margin of victory. What is the standard error of the spread \(2\bar{X}-1\) in this case?

Instructions

• Use the sqrt function to calculate the standard error of the spread \(2\bar{X}-1\).

Answer:

rr # N represents the number of people polled N <- 25 # p represents the proportion of Democratic voters p <- 0.45 # Calculate the standard error of the spread. Print this value to the console. 2 * sqrt(p * (1-p) / N)

[1] 0.1989975

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Exercise 10. Sample size

So far we have said that the difference between the proportion of Democratic voters and Republican voters is about 10% and that the standard error of this spread is about 0.2 when N=25. Select the statement that explains why this sample size is sufficient or not.

Answer:

This sample size is too small because the standard error is larger than the spread.

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