Homework 8: Sample Sizes and Confidence Intervals¶

Helpful Resource:

• Python Reference: Cheat sheet of helpful array & table methods used in Data 8!

Recommended Readings:

• Estimation

• Why the Mean Matters

Please complete this notebook by filling in the cells provided. Before you begin, execute the cell below to setup the notebook by importing some helpful libraries. Each time you start your server, you will need to execute this cell again.

Throughout this homework and all future ones, please be sure to not re-assign variables throughout the notebook! For example, if you use max_temperature in your answer to one question, do not reassign it later on. Otherwise, you will fail tests that you thought you were passing previously!

Deadline:

This assignment is due Wednesday, 4/8 at 11:00am PT. Submissions after this time will be accepted for 24 hours and will incur a 20% penalty. Any submissions later than this 24 hour period will not be accepted unless an extension has been granted as per the syllabus page. Turn it in by Tuesday, 4/7 at 11:00am PT for 5 extra credit points.

Directly sharing answers is not okay, but discussing problems with the course staff or with other students is encouraged. Refer to the syllabus page to learn more about how to learn cooperatively.

You should start early so that you have time to get help if you’re stuck. Office hours are held Monday through Friday in Warren Hall 101B or online. The office hours schedule appears here.

1. Bounding the Tail of a Distribution¶

A community has an average age of 45 years with a standard deviation of 5 years. We do not know how the ages are distributed.

In each part below, fill in the blank with a percent that makes the statement true without further assumptions, and explain your answer.

Note: Do not round your answers. Give the best answer that is possible with the information given.

Please review Section 14.2 and Section 14.2.5 of the textbook before proceeding with this section. You will be able to understand and solve the problems more efficiently!

Question 1.1. Answer the following statement: At least _______% of the people are between 25 and 65 years old. Assign your answer (a number between 0 and 100 without the % symbol) to the variable q1_1_percentage.

Next, select all valid reasons why you chose your answer, and assign them to the array q1_1_reasoning.

1. We must use Chebyshev’s inequality to create a bound on the percentage of people between 25 and 65 years old because we cannot make any assumptions about the distribution of ages within the community.

2. We can use what we know about the Normal distribution to create a bound on the percentage of people between 25 and 65 years old because we can assume the distribution of ages within the community are normally distributed.

3. 25 is 4 SDs below the mean and 65 is 4 SDs above the mean, so our bound is in the range between +/- 4 SDs from the mean.

4. 25 is 2 SDs below the mean and 65 is 2 SDs above the mean, so our bound is in the range between +/- 2 SDs from the mean.

(6 Points)

Question 1.2. Answer the following statement: At most _______% of the people have ages that are not in the range 25 years to 65 years. Assign your answer (a number between 0 and 100 without the % symbol) to the variable q1_2_percentage.

Next, explain your answer by assigning one of the following statements to the variable q1_2_reasoning. Hint: Consider what relationship exists between your answer to 1.1 and 1.2, and recall the percentage value you stored in q1_1_percentage!

1. Since we know that at least q1_1_percentage% of the people are between 25 and 65 years old, we subtract that from 100% to determine the maximum percentage of people that are not in this range.

2. Since we know that at least q1_1_percentage% of the people are between 25 and 65 years old, we divide by 2 to determine the maximum percentage of people that are not in this range.

3. Since we know that at least q1_1_percentage% of the people are between 25 and 65 years old, we subtract that from 100% and divide the difference by 2 to determine the maximum percentage of people that are not in this range.

(6 Points)

Question 1.3. Answer the following statement: At most _______% of the people are more than 65 years old. Assign your answer (a number between 0 and 100 without the % symbol) to the variable q1_3_percentage.

Next, explain your answer by assigning one of the following statements to the variable q1_3_reasoning.

Hint: If you’re stuck, try thinking about what the distribution may look like in this case.

1. Since the distribution is symmetric around the mean, at most half of the people outside the 25 to 65 years range must be more than 65 years old.

2. Since we do not know anything about the distribution, at most half of the people outside the 25 to 65 years range must be more than 65 years old.

3. Since the distribution is symmetric around the mean, the maximum possible percentage of people more than 65 years old is the same as the maximum amount of people that are not in the 25 to 65 years range.

4. Since we do not know anything about the distribution, the maximum possible percentage of people more than 65 years old is the same as the maximum amount of people that are not in the 25 to 65 years range.

(6 Points)

2. Sample Size and Confidence Level¶

A data science class at the large Data 8 University wants to estimate the percent of Facebook users among students at the school. To do this, they need to take a random sample of students. You can assume that their method of sampling is equivalent to drawing at random with replacement from students at the school.

Please review Section 14.6 of the textbook before proceeding with this section. There is a helpful formula that will help you solve the problems!

Question 2.1. Assign smallest to the smallest number of students they should sample to ensure that a 95% confidence interval for the parameter has a width of no more than 6% from left end to right end. (6 points)

Hint: How can our data be represented to show if a student in the sample is a Facebook user or not? Given this, what assumptions can we make for the SD of the population? Section 14.6 might be helpful!

Note: The ceil function will round up a float to the next highest integer. While your calculations for the smallest possible sample size may result in a float, a sample size must always be an integer. Write your calculations for the smallest possible sample size inside of ceil(...) to ensure that smallest is assigned to the smallest integer sample size that will satisfy our width requirements.

Question 2.2. Suppose the data science class decides to construct a 90% confidence interval instead of a 95% confidence interval, but they still require that the width of the interval is no more than 6% from left end to right end. Will they need the same sample size as in 2.1? Assign sample_size_answer to the correct answer. (6 Points)

1. Yes, they must use the same sample size, because the maximum width of the confidence interval has not changed.

2. No, a smaller sample size will work. A 90% confidence interval spans fewer standard deviations than a 95% confidence interval, so a smaller sample size can achieve the same maximum width of 6%.

3. No, they will need a bigger sample. A 90% confidence interval spans more standard deviations than a 95% confidence interval, so a larger sample size is needed to achieve the same maximum width of 6%.

Question 2.3. The professor tells the class that a 90% confidence interval for the parameter is constructed exactly like a 95% confidence interval, except that you have to go only 1.65 SDs on either side of the estimate (±1.65) instead of 2 SDs on either side (±2). Assign smallest_num to the smallest number of students they should sample to ensure that a 90% confidence interval for the parameter has a width of no more than 6% from left end to right end. (6 points)

Note: The ceil function will round up a float to the next highest integer. While your calculations for the smallest possible sample size may result in a float, a sample size must always be an integer. Write your calculations for the smallest possible sample size inside of ceil(...) to ensure that smallest_num is assigned to the smallest integer sample size that will satisfy our width requirements.

For this next exercise, please consult Section 14.3.4 of the textbook for similar examples.

Richard and Noah are curious about how the professor came up with the value 1.65 in Question 2.3. The professor says he ran the following two code cells. The first one calls the datascience library function plot_normal_cdf, which displays the proportion that is at most the specified number of SDs above average under the normal curve plotted with standard units on the horizontal axis. You can find the documentation here.

Note: The acronym cdf stands for cumulative distribution function. It measures the proportion to the left of a specified point under a probability histogram.

To run the second cell, the professor had to first import a Python library for probability and statistics:

Then he used the norm.cdf method in the library to find the gold proportion above.

This means that roughly 95% of our data lies to the left of +1.65 SDs from the mean (the shaded area in yellow above).

Note: You do not need to understand how the scipy library or how to use the method yourself.

Question 2.4. The cell above shows that in a normal distribution, about 95% of the data is less than or equal to 1.65 SDs above average. Therefore, what can we say about the right number of SDs to use when constructing a 90% confidence interval? Select all valid statements, and assign your answers to the array sd_answers. (6 Points)

1. A 90% confidence interval should be contained in about 1.65 SDs above and below the center because this captures the middle 90% of a normal distribution.

2. A 90% confidence interval should be contained in about 2 SDs above and below the center because it is the standard cutoff for normal distributions.

3. A 90% confidence interval should be contained in about 0.90 SDs above and below the center because the confidence level is 90%.

4. A 90% confidence interval should cover from the left end of the distribution up to +1.65 SDs above the center, since 95% of values are below that point.

5. Because the normal distribution is symmetric, if about 5% of values are above +1.65 SDs, then about 5% are below −1.65 SDs, so a 90% confidence interval uses ±1.65 SDs.

6. Because the normal distribution is symmetric, if about 5% of values are above +2 SDs, then about 5% are below −2 SDs, so a 90% confidence interval uses ±2 SDs.

Question 2.5. The cell above shows that the proportion that is at most 2.33 SDs above average in a normal distribution is 99%. Assign option to the right option to fill in the blank: (6 points)

If you start at the estimate and go 2.33 SDs on either side, then you will get a _______% confidence interval for the parameter.

1. 99.5

2. 99

3. 98.5

4. 98

Note: option should be assigned to one of 1, 2, 3, or 4 depending on which answer is correct.

3. Polling and the Normal Distribution¶

Marissa and Soumyadeep are a statistical consultants, and they are working for a group that supports Proposition 68 (which would mandate labeling of all horizontal and vertical axes, unrelated to any real California proposition) called Yes on 68. They want to know how many Californians will vote for the proposition.

Marissa polls a random sample of all California voters, and she finds that 210 of the 400 sampled voters will vote in favor of the proposition. We have provided a table for you below which has 3 columns: the first two columns are identical to sample. The third column contains the proportion of total voters that chose each option.

Question 3.1. Marissa wants to use 10,000 bootstrap resamples to compute a confidence interval for the proportion of all California voters who will vote Yes.

Fill in the next cell to simulate an empirical distribution of Yes proportions. Use bootstrap resampling to simulate 10,000 election outcomes, and assign resample_yes_proportions to contain the Yes proportion of each bootstrap resample. Then, visualize resample_yes_proportions with a histogram. You should see a bell shaped histogram centered near the proportion of Yes in the original sample. (6 points)

Hint: sample_proportions may be useful here!

Question 3.2. Consider the histogram of bootstrap resampled Yes proportions above. Which of the following statements about this distribution and the Central Limit Theorem are true? Assign your answers to the array clt_answers. We recommend reviewing 14.4 for a refresher on the CLT. (6 points)

1. If we think of Yes votes as 1s and No votes as 0s, each resampled Yes proportion is the mean of a bootstrap resample of 1s and 0s taken with replacement from the original sample, so the Central Limit Theorem applies when the sample size is large.

2. Because the sample size of 400 voters is large, the Central Limit Theorem tells us that the bootstrap distribution of the sample Yes proportions will be approximately normal, meaning it’s bell-shaped and symmetric.

3. The Central Limit Theorem applies only when the population itself follows a normal distribution.

4. The bootstrap distribution is centered around the proportion of Yes votes in the original sample (210/400).

5. The boostrap distribution should be centered at 0.5, since there are only two possible outcomes (Yes or No).

In a population whose members are represented as either a 0 or 1, there is a simple formula for the standard deviation of that population:

(Figuring out this formula, starting from the definition of the standard deviation, is a fun exercise for those who enjoy algebra.)

Question 3.3. Using only the Central Limit Theorem and the numbers of Yes and No voters in our sample of 400, algebraically compute the predicted standard deviation of the resample_yes_proportions array. Assign this number to approximate_sd. Do not access the data in resample_yes_proportions in any way. (6 points)

Remember that the standard deviation of the sample means can be computed from the population SD and the size of the sample (the formula above might be helpful). If we do not know the population SD, we can use the sample SD as a reasonable approximation in its place.

Note: Section 14.5.1 of the textbook may be helpful.

Question 3.4. Compute the standard deviation of the array resample_yes_proportions, which will act as an approximation to the true SD of the possible sample proportions. This will help verify whether your answer to question 3.3 is approximately correct. (6 points)

Question 3.5. Again, without accessing resample_yes_proportions in any way, compute an approximate 95% confidence interval for the proportion of Yes voters in California. (6 points)

The cell below draws your interval as a red bar below the histogram of resample_yes_proportions; use that to verify that your answer looks right.

Hint: How many SDs corresponds to 95% of the distribution promised by the CLT? Recall the discussion in the textbook here.

Hint: The approximate_sd variable you previously defined may be helpful!

Your confidence interval should overlap the number 0.5. That means we can’t be very sure whether Proposition 68 is winning, even though the sample Yes proportion is a bit above 0.5.

The Yes on 68 campaign really needs to know whether they’re winning. It’s impossible to be absolutely sure without polling the whole population, but they’d be okay if the standard deviation of the sample mean were only 0.005. They ask Marissa to run a new poll with a sample size that’s large enough to achieve that. (Polling is expensive, so the sample also shouldn’t be bigger than necessary.)

Marissa consults Chapter 14 of the textbook. Instead of making the conservative assumption that the population standard deviation is 0.5 (coding Yes voters as 1 and No voters as 0), she decides to assume that it’s equal to the standard deviation of the sample,

Under that assumption, Marissa decides that a sample size of 9,975 would suffice.

Does Marissa’s sample size achieve the desired standard deviation of sample means? What SD would you achieve with a smaller sample size? A higher sample size?

Question 3.6. To explore this, first compute the SD of sample means obtained by using Marissa’s sample size and assign it to marissa_sample_mean_sd. (6 points)

Question 3.7. Next, compute the SD of sample means that you would get from a smaller sample size. Ideally, you should pick a number that is significantly smaller, but any sample size smaller than Marissa’s will do. (5 points)

Question 3.8. Finally, compute the SD of sample means that you would get from a larger sample size. Here, a number that is significantly larger would make any difference more obvious, but any sample size larger than Marissa’s will do. (5 points)

Question 3.9. Based off of this, was Marissa’s sample size approximately the minimum sufficient sample, given her assumption that the sample SD is the same as the population SD? Assign min_sufficient to True if 9,975 was indeed approximately the minimum sufficient sample, and False if it wasn’t. (6 points)

You’re done with Homework 8!

Important submission steps:

1. Run the tests and verify that they all pass.

2. Choose Save Notebook from the File menu, then run the final cell.

3. Click the link to download the zip file.

4. Go to Pensive and submit the zip file to the corresponding assignment. The name of this assignment is “HW 08 Autograder”.

It is your responsibility to make sure your work is saved before running the last cell.

Pets of Data 8¶

Biscuit and Sandie are proud of you for completing the assignment!

Congrats on finishing Homework 8!

To double-check your work, the cell below will rerun all of the autograder tests.

Submission¶

Make sure you have run all cells in your notebook in order before running the cell below, so that all images/graphs appear in the output. The cell below will generate a zip file for you to submit. Please save before exporting!