Discussion 7: Standardization and the Normal Distribution

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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

Hector earned a 77 on an exam where the mean was 70 and the standard deviation was 5.
Clara earned an 80 on an exam where the mean was 75 and the standard deviation was 3.
Vivek earned an 83 on an exam where the mean was a 75 and the standard deviation was 6.

Rank these three students in ascending order of their exam performance relative to their classmates.

Hector, Clara, Vivek
Vivek, Hector, Clara
Clara, Hector, Vivek
Vivek, Clara, Hector

Problem 2

The data visualization below shows all Olympic gold medals for women’s gymnastics, broken down by the age of the gymnast.

Based on this data, rank the following three quantities in ascending order: the median age at which gold medals are earned, the mean age at which gold medals are earned, the standard deviation of the age at which gold medals are earned.

mean, median, SD
median, mean, SD
SD, mean, median
SD, median, mean

Problem 3

Among all Costco members in San Diego, the average monthly spending in October 2023 was $350 with a standard deviation of $40.

Problem 3.1

The amount Ciro spent at Costco in October 2023 was -1.5 in standard units. What is this amount in dollars? Give your answer as an integer.

Problem 3.2

What is the minimum possible percentage of San Diego members that spent between $250 and $450 in October 2023?

Problem 3.3

Now, suppose we’re given that the distribution of monthly spending in October 2023 for all San Diego members is roughly normal. Given this fact, fill in the blanks:

"In October 2023, 95% of San Diego members spent between __(m)__ dollars and __(n)__ dollars."

What are m and n? Give your answers as integers rounded to the nearest multiple of 10.

Problem 4

Researchers from the San Diego Zoo, located within Balboa Park, collected physical measurements of several species of penguins in a region of Antarctica.

One piece of information they tracked for each of 330 penguins was its mass in grams. The average penguin mass is 4200 grams, and the standard deviation is 840 grams.

Problem 4.1

Consider the histogram of mass below.

Select the true statement below.

The median mass of penguins is larger than the average mass of penguins
The median mass of penguins is roughly equal to the average mass of penguins (within 50 grams)
The median mass of penguins is less than the average mass of penguins
It is impossible to determine the relationship between the median and average mass of penguins just by looking at the above histogram

Problem 4.2

For your convenience, we show the histogram of mass again below.

Recall, there are 330 penguins in our dataset. Their average mass is 4200 grams, and the standard deviation of mass is 840 grams.

Per Chebyshev’s inequality, at least what percentage of penguins have a mass between 3276 grams and 5124 grams? Input your answer as a percentage between 0 and 100, without the % symbol. Round to three decimal places.

Problem 4.3

Per Chebyshev’s inequality, at least what percentage of penguins have a mass between 1680 grams and 5880 grams?

Problem 4.4

The distribution of mass in grams is not roughly normal. Is the distribution of mass in standard units roughly normal?

Yes
No
Impossible to tell

Problem 4.5

Suppose boot_means is an array of the resampled means. Fill in the blanks below so that [left, right] is a 68% confidence interval for the true mean mass of penguins.

left = np.percentile(boot_means, __(a)__)
right = np.percentile(boot_means, __(b)__)
[left, right]

What goes in blank (a)? What goes in blank (b)?

Problem 4.6

Which of the following is a correct interpretation of this confidence interval? Select all that apply.

There is an approximately 68% chance that mean weight of all penguins in Antarctica falls within the bounds of this confidence interval.
Approximately 68% of penguin weights in our sample fall within the bounds of this confidence interval.
Approximately 68% of penguin weights in the population fall within the bounds of this interval.
If we created many confidence intervals using the same method, approximately 68% of them would contain the mean weight of all penguins in Antarctica.
None of the above

Problem 5

An IKEA chair designer is experimenting with some new ideas for armchair designs. She has the idea of making the arm rests shaped like bell curves, or normal distributions. A cross-section of the armchair design is shown below.

This was created by taking the portion of the standard normal distribution from z=-4 to z=4 and adjoining two copies of it, one centered at z=0 and the other centered at z=8. Let’s call this shape the armchair curve.

Since the area under the standard normal curve from z=-4 to z=4 is approximately 1, the total area under the armchair curve is approximately 2.

Complete the implementation of the two functions below:

area_left_of(z) should return the area under the armchair curve to the left of z, assuming -4 <= z <= 12, and
area_between(x, y) should return the area under the armchair curve between x and y, assuming -4 <= x <= y <= 12.

import scipy

def area_left_of(z):
    '''Returns the area under the armchair curve to the left of z.
       Assume -4 <= z <= 12'''
    if ___(a)___: 
        return ___(b)___ 
    return scipy.stats.norm.cdf(z)

def area_between(x, y):
    '''Returns the area under the armchair curve between x and y. 
    Assume -4 <= x <= y <= 12.'''
    return ___(c)___

Problem 5.1

What goes in blank (a)?

Problem 5.2

What goes in blank (b)?

Problem 5.3

What goes in blank (c)?

Problem 6

Suppose you have correctly implemented the function area_between(x, y) so that it returns the area under the armchair curve between x and y, assuming the inputs satisfy -4 <= x <= y <= 12.

Note: You can still do this question, even if you didn’t know how to do the previous one.

Problem 6.1

What is the approximate value of area_between(-2, 10)?

1.9
1.95
1.975
2

Problem 6.2

What is the approximate value of area_between(0.37, 8.37)?

0.68
0.95
1
1.5

Problem 7

Beneath Gringotts Wizarding Bank, enchanted mine carts transport wizards through a complex underground railway on the way to their bank vault.

During one section of the journey to Harry’s vault, the track follows the shape of a normal curve, with a peak at x = 50 and a standard deviation of 20.

Problem 7.1

A ferocious dragon, who lives under this section of the railway, is equally likely to be located anywhere within this region. What is the probability that the dragon is located in a position with x \leq 10 or x \geq 80? Select all that apply.

1 - (scipy.stats.norm.cdf(1.5) - scipy.stats.norm.cdf(-2))
2 * scipy.stats.norm.cdf(1.75)
scipy.stats.norm.cdf(-2) + scipy.stats.norm.cdf(-1.5)
0.95
None of the above.

Problem 7.2

Harry wants to know where, in this section of the track, the cart’s height is changing the fastest. He knows from his earlier public school education that the height changes the fastest at the inflection points of a normal distribution. Where are the inflection points in this section of the track?

x = 50
x = 20 and x = 80
x = 30 and x = 70
x = 0 and x = 100

Problem 7.3

Next, consider a different region of the track, where the shape follows some arbitrary distribution with mean 130 and standard deviation 30. We don’t have any information about the shape of the distribution, so it is not necessarily normal.

What is the minimum proportion of area under this section of the track within the range 100 \leq x \leq 190?

0.77
0.55
0.38
0.00