Tesla (a car company) is developing a new vehicle that is very expensive. The marketing team is working on how to advertise this niche product to their market. Since the car is so expensive, their population is very small so they are able to interview all people in the population. The team is interested in whether the end user will buy the product or not. Let Y be an indicator variable, indicating whether or not the person would purchase the car. If Y=1 then the person said "Yes, they would purchase the car", and if Y=0 then they would "Not purchase the car".

The data is below

Y = {1, 1, 1, 1, 0, 1, 1, 1, 1}

What is the variance of Y?

(Note: you must use the formula in Ch 1 of the Text, or in the "Population & Sample Introduction" Video)

Submit your answer as a number rounded to 3 decimals. So 40.12357 would be input as 40.124

Question

Tesla (a car company) is developing a new vehicle that is very expensive. The marketing team is working on how to advertise this niche product to their market. Since the car is so expensive, their population is very small so they are able to interview all people in the population. The team is interested in whether the end user will buy the product or not. Let Y be an indicator variable, indicating whether or not the person would purchase the car. If Y=1 then the person said "Yes, they would purchase the car", and if Y=0 then they would "Not purchase the car".

The data is below

Y = {1, 1, 1, 1, 0, 1, 1, 1, 1}

What is the variance of Y?

(Note: you must use the formula in Ch 1 of the Text, or in the "Population & Sample Introduction" Video)

Submit your answer as a number rounded to 3 decimals. So 40.12357 would be input as 40.124

BlackTom AI · Accepted Answer

We start by identifying the data: Y is a binary indicator taking values 1 (Yes, purchase) and 0 (Not purchase). There are 9 observations: {1, 1, 1, 1, 0, 1, 1, 1, 1}.

Step 1: Compute the sample proportion of 1s (the sample mean for a Bernoulli variable).
p_hat = number of 1s / total observations = 8 / 9 ≈ 0.8889.

Step 2: Compute the Bernoulli variance using the standard population formula.
Population variance for a Bernoulli(p) is p(1 - p).
Here, p = p_hat ≈ 0.8889, so p(1 - p) ≈ 0.8889 * 0.1111 ≈ 0.098765.

Step 3: Since the instruction refers to using the formula from Chapter 1 or the Population & Sample Introduction video, the dataset asks for the variance estimator that uses the sample (i.e., the unbiased sample variance for a Bernoulli). The unbiased estimator uses a denominator of (n - 1):
S^2 = [sum (Y_i - Y_bar)^2] / (n - 1).
For a Bernoulli sample, this simplifies to S^2 = p_hat(1 - p_hat) * n/(n - 1).

Step 4: Apply the correction factor
n = 9, so n/(n - 1) = 9/8 = 1.125.
S^2 = 0.098765... * 1.125 = 0.111111...

Step 5: Round to three decimals as requested.
0.111111... rounds to 0.111.

Summary:
- Count of 1s = 8, n = 9, p_hat = 8/9.
- Unbiased sample variance for Bernoulli = p_hat(1 - p_hat) * n/(n - 1) = 0.111111...
- Rounded value = 0.111.

STA304H1 S Module 3 Practice Problems

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