Discussion 9: The Central Limit Theorem

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These problems are taken from past quizzes and exams. Work on them on paper, since the quizzes and exams you take in this course will also be on paper.

We encourage you to complete these problems during discussion section. Solutions will be made available after all discussion sections have concluded. You don’t need to submit your answers anywhere.

Note: We do not plan to cover all of these problems during the discussion section; the problems we don’t cover can be used for extra practice.

Problem 1

Oren has a random sample of 200 dog prices in an array called oren. He has also bootstrapped his sample 1,000 times and stored the mean of each resample in an array called boots.

In this question, assume that the following code has run:

a = np.mean(oren)
b = np.std(oren)
c = len(oren)


Problem 1.1

What expression best estimates the population’s standard deviation?

Answer: b

The function np.std directly calculated the standard deviation of array oren. Even though oren is sample of the population, its standard deviation is still a pretty good estimate for the standard deviation of the population because it is a random sample. The other options don’t really make sense in this context.

Difficulty: ⭐️⭐️⭐️

The average score on this problem was 57%.


Problem 1.2

Which expression best estimates the mean of boots?

Answer: a

Note that a is equal to the mean of oren, which is a pretty good estimator of the mean of the overall population as well as the mean of the distribution of sample means. The other options don’t really make sense in this context.

Difficulty: ⭐️⭐️

The average score on this problem was 89%.


Problem 1.3

What expression best estimates the standard deviation of boots?

Answer: b / np.sqrt(c)

Note that we can use the Central Limit Theorem for this problem which states that the standard deviation (SD) of the distribution of sample means is equal to (population SD) / np.sqrt(sample size). Since the SD of the sample is also the SD of the population in this case, we can plug our variables in to see that b / np.sqrt(c) is the answer.

Difficulty: ⭐️

The average score on this problem was 91%.


Problem 1.4

What is the dog price of $560 in standard units?

Answer: (560 - a) / b

To convert a value to standard units, we take the value, subtract the mean from it, and divide by SD. In this case that is (560 - a) / b, because a is the mean of our dog prices sample array and b is the SD of the dog prices sample array.

Difficulty: ⭐️⭐️

The average score on this problem was 80%.


Problem 1.5

The distribution of boots is normal because of the Central Limit Theorem.

Answer: True

True. The central limit theorem states that if you have a population and you take a sufficiently large number of random samples from the population, then the distribution of the sample means will be approximately normally distributed.

Difficulty: ⭐️

The average score on this problem was 91%.


Problem 1.6

If Oren’s sample was 400 dogs instead of 200, the standard deviation of boots will…

Answer: Decrease by a factor of \sqrt{2}

Recall that the central limit theorem states that the STD of the sample distribution is equal to (population STD) / np.sqrt(sample size). So if we increase the sample size by a factor of 2, the STD of the sample distribution will decrease by a factor of \sqrt{2}.

Difficulty: ⭐️⭐️

The average score on this problem was 80%.


Problem 1.7

If Oren took 4000 bootstrap resamples instead of 1000, the standard deviation of boots will…

Answer: None of the above

Again, from our formula given by the central limit theorem, the sample STD doesn’t depend on the number of bootstrap resamples so long as it’s “sufficiently large”. Thus increasing our bootstrap sample from 1000 to 4000 will have no effect on the std of boots

Difficulty: ⭐️⭐️⭐️

The average score on this problem was 74%.


Problem 1.8

Write one line of code that evaluates to the right endpoint of a 92% CLT-Based confidence interval for the mean dog price. The following expressions may help:

stats.norm.cdf(1.75) # => 0.96
stats.norm.cdf(1.4)  # => 0.92

Answer: a + 1.75 * b / np.sqrt(c)

Recall that a 92% confidence interval means an interval that consists of the middle 92% of the distribution. In other words, we want to “chop” off 4% from either end of the ditribution. Thus to get the right endpoint, we want the value corresponding to the 96th percentile in the mean dog price distribution, or mean + 1.75 * (SD of population / np.sqrt(sample size) or a + 1.75 * b / np.sqrt(c) (we divide by np.sqrt(c) due to the central limit theorem). Note that the second line of information that was given stats.norm.cdf(1.4) is irrelavant to this particular problem.

Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 48%.


Problem 2

From a population with mean 500 and standard deviation 50, you collect a sample of size 100. The sample has mean 400 and standard deviation 40. You bootstrap this sample 10,000 times, collecting 10,000 resample means.


Problem 2.1

Which of the following is the most accurate description of the mean of the distribution of the 10,000 bootstrapped means?

Answer: The mean will be approximately equal to 400.

The distribution of bootstrapped means’ mean will be approximately 400 since that is the mean of the sample and bootstrapping is taking many samples of the original sample. The mean will not be exactly 400 do to some randomness though it will be very close.

Difficulty: ⭐️⭐️⭐️

The average score on this problem was 54%.


Problem 2.2

Which of the following is closest to the standard deviation of the distribution of the 10,000 bootstrapped means?

Answer: 4

To find the standard deviation of the distribution, we can take the sample standard deviation S divided by the square root of the sample size. From plugging in, we get 40 / 10 = 4.

Difficulty: ⭐️⭐️⭐️

The average score on this problem was 51%.


Problem 3

Suppose you draw a sample of size 100 from a population with mean 50 and standard deviation 15. What is the probability that your sample has a mean between 50 and 53? Input the probability below, as a number between 0 and 1, rounded to two decimal places.

Answer: 0.48

This problem is testing our understanding of the Central Limit Theorem and normal distributions. Recall, the Central Limit Theorem tells us that the distribution of the sample mean is roughly normal, with the following characteristics:

\begin{align*} \text{Mean of Distribution of Possible Sample Means} &= \text{Population Mean} = 50 \\ \text{SD of Distribution of Possible Sample Means} &= \frac{\text{Population SD}}{\sqrt{\text{Sample Size}}} = \frac{15}{\sqrt{100}} = 1.5 \end{align*}

Given this information, it may be easier to express the problem as “We draw a value from a normal distribution with mean 50 and SD 1.5. What is the probability that the value is between 50 and 53?” Note that this probability is equal to the proportion of values between 50 and 53 in a normal distribution whose mean is 50 and 1.5 (since probabilities can be thought of as proportions).

In class, we typically worked with the standard normal distribution, in which the mean was 0, the SD was 1, and the x-axis represented values in standard units. Let’s convert the quantities of interest in this problem to standard units, keeping in mind that the mean and SD we’re using now are the mean and SD of the distribution of possible sample means, not of the population.

  • 50 converted to standard units is \frac{50 - \text{mean}}{\text{SD}} = \frac{50 - 50}{1.5} = 0 (no calculation was necessary – 0 in standard units is equal to the mean in original units).
  • 53 converted to standard units is \frac{53 - \text{mean}}{\text{SD}} = \frac{53 - 50}{1.5} = 2.

Now, our problem boils down to finding the proportion of values in a standard normal distribution that are between 0 and 2, or the proportion of values in a normal distribution that are in the interval [\text{mean}, \text{mean} + 2 \text{ SDs}].

From class, we know that in a normal distribution, roughly 95% of values are within 2 standard deviations of the mean, i.e. the proportion of values in the interval [\text{mean} - 2 \text{ SDs}, \text{mean} + 2 \text{ SDs}] is 0.95.

Since the normal distribution is symmetric about the mean, half of the values in this interval are to the right of the mean, and half are to the left. This means that the proportion of values in the interval [\text{mean}, \text{mean} + 2 \text{ SDs}] is \frac{0.95}{2} = 0.475, which rounds to 0.48, and thus the desired result is 0.48.

Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 48%.

Problem 4

The DataFrame apps contains application data for a random sample of 1,000 applicants for a particular credit card from the 1990s. The "age" column contains the applicants’ ages, in years, to the nearest twelfth of a year.

The credit card company that owns the data in apps, BruinCard, has decided not to give us access to the entire apps DataFrame, but instead just a random sample of 100 rows of apps called hundred_apps.

We are interested in estimating the mean age of all applicants in apps given only the data in hundred_apps. The ages in hundred_apps have a mean of 35 and a standard deviation of 10.


Problem 4.1

Give the endpoints of the CLT-based 95% confidence interval for the mean age of all applicants in apps, based on the data in hundred_apps.

Answer: Left endpoint = 33, Right endpoint = 37

According to the Central Limit Theorem, the standard deviation of the distribution of the sample mean is \frac{\text{sample SD}}{\sqrt{\text{sample size}}} = \frac{10}{\sqrt{100}} = 1. Then using the fact that the distribution of the sample mean is roughly normal, since 95% of the area of a normal curve falls within two standard deviations of the mean, we can find the endpoints of the 95% CLT-based confidence interval as 35 - 2 = 33 and 35 + 2 = 37.

We can think of this as using the formula below: \left[\text{sample mean} - 2\cdot \frac{\text{sample SD}}{\sqrt{\text{sample size}}}, \: \text{sample mean} + 2\cdot \frac{\text{sample SD}}{\sqrt{\text{sample size}}} \right]. Plugging in the appropriate quantities yields [35 - 2\cdot\frac{10}{\sqrt{100}}, 35 - 2\cdot\frac{10}{\sqrt{100}}] = [33, 37].

Difficulty: ⭐️⭐️⭐️

The average score on this problem was 67%.


Problem 4.2

BruinCard reinstates our access to apps so that we can now easily extract information about the ages of all applicants. We determine that, just like in hundred_apps, the ages in apps have a mean of 35 and a standard deviation of 10. This raises the question of how other samples of 100 rows of apps would have turned out, so we compute 10,000 sample means as follows.

    sample_means = np.array([])
    for i in np.arange(10000):
        sample_mean = apps.sample(100, replace=True).get("age").mean()
        sample_means = np.append(sample_means, sample_mean)

Which of the following three visualizations best depict the distribution of sample_means?


Answer: Option 1

As we found in the previous part, the distribution of the sample mean should have a standard deviation of 1. We also know it should be centered at the mean of our sample, at 35, but since all the options are centered here, that’s not too helpful. Only Option 1, however, has a standard deviation of 1. Remember, we can approximate the standard deviation of a normal curve as the distance between the mean and either of the inflection points. Only Option 1 looks like it has inflection points at 34 and 36, a distance of 1 from the mean of 35.

If you chose Option 2, you probably confused the standard deviation of our original sample, 10, with the standard deviation of the distribution of the sample mean, which comes from dividing that value by the square root of the sample size.

Difficulty: ⭐️⭐️⭐️

The average score on this problem was 57%.


Problem 4.3

Which of the following statements are guaranteed to be true? Select all that apply.

Answer: A CLT-based 90% confidence interval for the mean age of credit card applicants, based on the data in hundred_apps, would be narrower than the interval you gave in part (a).

Let’s analyze each of the options:

  • Option 1: We are not using bootstrapping to compute sample means since we are sampling from the apps DataFrame, which is our population here. If we were bootstrapping, we’d need to sample from our first sample, which is hundred_apps.

  • Option 2: We can’t be sure what the distribution of the ages of credit card applicants are. The Central Limit Theorem says that the distribution of sample_means is roughly normally distributed, but we know nothing about the population distribution.

  • Option 3: The CLT-based 95% confidence interval that we calculated in part (a) was computed as follows: \left[\text{sample mean} - 2\cdot \frac{\text{sample SD}}{\sqrt{\text{sample size}}}, \text{sample mean} + 2\cdot \frac{\text{sample SD}}{\sqrt{\text{sample size}}} \right] A CLT-based 90% confidence interval would be computed as \left[\text{sample mean} - z\cdot \frac{\text{sample SD}}{\sqrt{\text{sample size}}}, \text{sample mean} + z\cdot \frac{\text{sample SD}}{\sqrt{\text{sample size}}} \right] for some value of z less than 2. We know that 95% of the area of a normal curve is within two standard deviations of the mean, so to only pick up 90% of the area, we’d have to go slightly less than 2 standard deviations away. This means the 90% confidence interval will be narrower than the 95% confidence interval.

  • Option 4: The left endpoint of the interval from part (a) was calculated using the Central Limit Theorem, whereas using np.percentile(sample_means, 2.5) is calculated empirically, using the data in sample_means. Empirically calculating a confidence interval doesn’t necessarily always give the exact same endpoints as using the Central Limit Theorem, but it should give you values close to those endpoints. These values are likely very similar but they are not guaranteed to be the same. One way to see this is that if we ran the code to generate sample_means again, we’d probably get a different value for np.percentile(sample_means, 2.5).

  • Option 5: The key observation is that if we used the data in hundred_apps to create 1,000 CLT-based 95% confidence intervals for the mean age of applicants in apps, all of these intervals would be exactly the same. Given a sample, there is only one CLT-based 95% confidence interval associated with it. In our case, given the sample hundred_apps, the one and only CLT-based 95% confidence interval based on this sample is the one we found in part (a). Therefore if we generated 1,000 of these intervals, either they would all contain the parameter or none of them would. In order for a statement like the one here to be true, we would need to collect 1,000 different samples, and calculate a confidence interval from each one.

Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 49%.


Problem 5

You need to estimate the proportion of American adults who want to be vaccinated against Covid-19. You plan to survey a random sample of American adults, and use the proportion of adults in your sample who want to be vaccinated as your estimate for the true proportion in the population. Your estimate must be within 0.04 of the true proportion, 95% of the time. Using the fact that the standard deviation of any dataset of 0’s and 1’s is no more than 0.5, calculate the minimum number of people you would need to survey. Input your answer below, as an integer.

Answer: 625

Note: Before reviewing these solutions, it’s highly recommended to revisit the lecture on “Choosing Sample Sizes,” since this problem follows the main example from that lecture almost exactly.

While this solution is long, keep in mind from the start that our goal is to solve for the smallest sample size necessary to create a confidence interval that achieves certain criteria.

The Central Limit Theorem tells us that the distribution of the sample mean is roughly normal, regardless of the distribution of the population from which the samples are drawn. At first, it may not be clear how the Central Limit Theorem is relevant, but remember that proportions are means too – for instance, the proportion of adults who want to be vaccinated is equal to the mean of a collection of 1s and 0s, where we have a 1 for each adult that wants to be vaccinated and a 0 for each adult who doesn’t want to be vaccinated. What this means (😉) is that the Central Limit Theorem applies to the distribution of the sample proportion, so we can use it here too.

Not only do we know that the distribution of sample proportions is roughly normal, but we know its mean and standard deviation, too:

\begin{align*} \text{Mean of Distribution of Possible Sample Means} &= \text{Population Mean} = \text{Population Proportion} \\ \text{SD of Distribution of Possible Sample Means} &= \frac{\text{Population SD}}{\sqrt{\text{Sample Size}}} \end{align*}

Using this information, we can create a 95% confidence interval for the population proportion, using the fact that in a normal distribution, roughly 95% of values are within 2 standard deviations of the mean:

\left[ \text{Population Proportion} - 2 \cdot \frac{\text{Population SD}}{\sqrt{\text{Sample Size}}}, \: \text{Population Proportion} + 2 \cdot \frac{\text{Population SD}}{\sqrt{\text{Sample Size}}} \right]

However, this interval depends on the population proportion (mean) and SD, which we don’t know. (If we did know these parameters, there would be no need to collect a sample!) Instead, we’ll use the sample proportion and SD as rough estimates:

\left[ \text{Sample Proportion} - 2 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}}, \: \text{Sample Proportion} + 2 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}} \right]

Note that the width of this interval – that is, its right endpoint minus its left endpoint – is: \text{width} = 4 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}}

In the problem, we’re told that we want our interval to be accurate to within 0.04, which is equivalent to wanting the width of our interval to be less than or equal to 0.08 (since the interval extends the same amount above and below the sample proportion). As such, we need to pick the smallest sample size necessary such that:

\text{width} = 4 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}} \leq 0.08

We can re-arrange the inequality above to solve for our sample’s size:

\begin{align*} 4 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}} &\leq 0.08 \\ \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}} &\leq 0.02 \\ \frac{1}{\sqrt{\text{Sample Size}}} &\leq \frac{0.02}{\text{Sample SD}} \\ \frac{\text{Sample SD}}{0.02} &\leq \sqrt{\text{Sample Size}} \\ \left( \frac{\text{Sample SD}}{0.02} \right)^2 &\leq \text{Sample Size} \end{align*}

All we now need to do is pick the smallest sample size that satisfies the above inequality. But there’s an issue – we don’t know what our sample SD is, because we haven’t collected our sample! Notice that in the inequality above, as the sample SD increases, so does the minimum necessary sample size. In order to ensure we don’t collect too small of a sample (which would result in the width of our confidence interval being larger than desired), we can use an upper bound for the SD of our sample. In the problem, we’re told that the largest possible SD of a sample of 0s and 1s is 0.5 – this means that if we replace our sample SD with 0.5, we will find a sample size such that the width of our confidence interval is guaranteed to be less than or equal to 0.08. This sample size may be larger than necessary, but that’s better than it being smaller than necessary.

By substituting 0.5 for the sample SD in the last inequality above, we get

\begin{align*} \left( \frac{\text{Sample SD}}{0.02} \right)^2 &\leq \text{Sample Size} \\\ \left( \frac{0.5}{0.02} \right)^2 &\leq \text{Sample Size} \\ 25^2 &\leq \text{Sample Size} \implies \text{Sample Size} \geq 625 \end{align*}

We need to pick the smallest possible sample size that is greater than or equal to 625; that’s just 625.

Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 40%.

Problem 6

It’s your first time playing a new game called Brunch Menu. The deck contains 96 cards, and each player will be dealt a hand of 9 cards. The goal of the game is to avoid having certain cards, called Rotten Egg cards, which come with a penalty at the end of the game. But you’re not sure how many of the 96 cards in the game are Rotten Egg cards. So you decide to use the Central Limit Theorem to estimate the proportion of Rotten Egg cards in the deck based on the 9 random cards you are dealt in your hand.


Problem 6.1

You are dealt 3 Rotten Egg cards in your hand of 9 cards. You then construct a CLT-based 95% confidence interval for the proportion of Rotten Egg cards in the deck based on this sample. Approximately, how wide is your confidence interval?

Choose the closest answer, and use the following facts:

Answer: \frac{17}{27}

A Central Limit Theorem-based 95% confidence interval for a population proportion is given by the following:

\left[ \text{Sample Proportion} - 2 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}}, \text{Sample Proportion} + 2 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}} \right]

Note that this interval uses the fact that (about) 95% of values in a normal distribution are within 2 standard deviations of the mean. It’s key to divide by \sqrt{\text{Sample Size}} when computing the standard deviation because the distribution that is roughly normal is the distribution of the sample mean (and hence, sample proportion), not the distribution of the sample itself.

The width of the above interval – that is, the right endpoint minus the left endpoint – is

\text{width} = 4 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}}

From the provided hint, we have that

\text{Sample SD} = \sqrt{(\text{Prop. of 0s}) \cdot (\text{Prop of 1s})} = \sqrt{\frac{3}{9} \cdot \frac{6}{9}} = \frac{\sqrt{18}}{9}

Then, since we know that the sample size is 9 and that \sqrt{18} is about \frac{17}{4}, we have

\text{width} = 4 \cdot \frac{\text{Sample SD}}{\sqrt{\text{Sample Size}}} = 4 \cdot \frac{\frac{\sqrt{18}}{9}}{\sqrt{9}} = 4 \cdot \frac{\sqrt{18}}{9 \cdot 3} = 4 \cdot \frac{\frac{17}{4}}{27} = \frac{17}{27}

Difficulty: ⭐️⭐️⭐️

The average score on this problem was 51%.


Problem 6.2

Which of the following are limitations of trying to use the Central Limit Theorem for this particular application? Select all that apply.

Answer: Options 1 and 2

Option 1: We use Central Limit Theorem (CLT) for large random samples, and a sample of 9 is considered to be very small. This makes it difficult to use CLT for this problem.

Option 2: Recall CLT happens when our sample is drawn with replacement. When we are handed nine cards we are never replacing cards back into our deck, which means that we are sampling without replacement.

Option 3: This is wrong because CLT states that a large sample is approximately a normal distribution even if the data itself is not normally distributed. This means it doesn’t matter if our data had not been normally distributed if we had a large enough sample we could use CLT.

Option 4: This is wrong because CLT does apply to the sample proportion distribution. Recall that proportions can be treated like means.

Difficulty: ⭐️⭐️

The average score on this problem was 77%.


Problem 7


Problem 7.1

Suppose you want to estimate the proportion of UCSD students that prefer sunny days over rainy days. You plan to survey 900 students, then construct a 95% confidence interval for this proportion. What is the widest possible width for the resulting confidence interval? Give your answer as a fully simplified fraction.

Answer: \frac{1}{15}

Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 38%.


Problem 7.2

If you decide to survey 450 students instead of 900 students for your sample, the maximum possible width of your 95% confidence interval would:

Answer: increase by less than double

Difficulty: ⭐️⭐️⭐️

The average score on this problem was 60%.


Problem 8

You want to estimate the proportion of DSC majors who have a Netflix subscription. To do so, you will survey a random sample of DSC majors and ask them whether they have a Netflix subscription. You will then create a 95% confidence interval for the proportion of “yes" answers in the population, based on the responses in your sample. You decide that your confidence interval should have a width of at most 0.10.


Problem 8.1

In order for your confidence interval to have a width of at most 0.10, the standard deviation of the distribution of the sample proportion must be at most T. What is T? Give your answer as an exact decimal.

Answer: 0.025

Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 46%.


Problem 8.2

Using the fact that the standard deviation of any dataset of 0s and 1s is no more than 0.5, calculate the minimum number of people you would need to survey so that the width of your confidence interval is at most 0.10. Give your answer as an integer.

Answer: 400

Difficulty: ⭐️⭐️

The average score on this problem was 81%.


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