# Spring 2024 Quiz 4

This quiz was administered in-person. It was closed-book and closed-note; students were not allowed to use the DSC 10 Reference Sheet. Students had 20 minutes to work on the quiz.

This quiz covered Lectures 17-19 of the Spring 2024 offering of DSC 10.

## Problem 1

### Problem 1.1

Suppose that the trees on UCSD’s campus have a mean height of 100 feet and a variance of 36 feet. If the height of a specific tree is 124 feet, what would its height be in standard units for this distribution? Simplify your answer.

##### Difficulty: ⭐️⭐️⭐️

The average score on this problem was 73%.

### Problem 1.2

Let A be the answer to the previous question. Choose the best interpretation of A.

• A randomly selected tree on UCSD’s campus has an A percent probability of being at least 124 feet tall.

• A 124 foot tree is A times taller than the average tree on UCSD’s campus.

• A 124 foot tree is A standard deviations taller than the average tree on UCSD’s campus.

• At least 95% of trees on UCSD’s campus have a height within A standard deviations of the mean height.

Answer: A 124 foot tree is A standard deviations taller than the average tree on UCSD’s campus.

##### Difficulty: ⭐️⭐️

The average score on this problem was 84%.

## Problem 2

You are told that scipy.stats.norm.cdf(-1.4) evaluates to 0.08075665923377107. Suppose you have a standard normal curve with mean at x=0 and standard deviation 1. What is the area under the curve from x=0 to x=1.4? Give your answer as a number rounded to 2 decimal places.

##### Difficulty: ⭐️⭐️⭐️

The average score on this problem was 57%.

## Problem 3

Suppose we measure the height in feet of a sample of trees on UCSD’s campus and use this sample to generate a 95% CLT-based confidence interval for the mean height of trees on campus. Let W be the width of this confidence interval.

If we instead were to measure the height of the same sample in inches, and again generate a 95% CLT-based confidence interval for the mean, what would be the width of this confidence interval in terms of W? There are 12 inches in 1 foot.

• \dfrac{W}{12}

• \dfrac{W}{\sqrt{12}}

• W

• 12W

• 144W

##### Difficulty: ⭐️⭐️⭐️

The average score on this problem was 63%.

## Problem 4

Which of the following quantities must be known in order to construct a CLT-based confidence interval for the population mean? Select all that apply.

• Shape of the population (normal or not)

• Shape of the sample (normal or not)

• Mean of the population

• Mean of the sample

• Standard deviation of the population

• Standard deviation of the sample

• Size of the population

• Size of the sample

Answer: Mean of the sample, Standard deviation of the sample, Size of the sample

##### Difficulty: ⭐️⭐️⭐️

The average score on this problem was 73%.

## Problem 5

We want to collect a sample of trees and use this sample to determine the proportion of all trees that are oak trees (a population parameter). We want to create a 95% confidence interval that is at most 0.04 wide. Which of the following inequalities should we use to find the smallest viable sample size we could collect?

• \text{sample size} \geq (4 * \frac{0.5}{0.04})^2

• \text{sample size} \geq (2 * \frac{0.5}{0.04})^2

• \text{sample size} \geq (4 * \frac{1}{0.04})^2

• \text{sample size} \geq (2 * \frac{1}{0.04})

Answer: \text{sample size} \geq (4 * \frac{0.5}{0.04})^2

##### Difficulty: ⭐️⭐️⭐️

The average score on this problem was 52%.

## Problem 6

Suppose that the trees on UCSD’s campus are 35% eucalyptus, 25% pine, and the remaining 40% some other variety. Write one line of code to simulate the act of randomly sampling 40 trees from this distribution, with replacement. Your code should output an array of length 3 where the elements represent the number of eucalyptus, pine, and other trees, respectively.

Answer: np.random.multinomial(40, [0.35, 0.25, 0.40])

##### Difficulty: ⭐️⭐️

The average score on this problem was 85%.