Why statistics quantifies uncertainty in SciPy - Challenge Your Understanding

Challenge - 5 Problems

❓ Predict Output

intermediate

Understanding Confidence Interval Calculation

What is the output of the following code that calculates a 95% confidence interval for a sample mean?

SciPy

import numpy as np
from scipy import stats

np.random.seed(0)
sample = np.random.normal(loc=50, scale=5, size=30)
mean = np.mean(sample)
sem = stats.sem(sample)
conf_int = stats.t.interval(0.95, len(sample)-1, loc=mean, scale=sem)
print(tuple(round(x, 2) for x in conf_int))

A (47.50, 52.66)

B (49.00, 51.00)

C (48.50, 51.50)

D (48.01, 52.15)

💡 Hint

Recall that the confidence interval depends on the sample mean, standard error, and t-distribution quantiles.

The code calculates the 95% confidence interval using the t-distribution for a sample of size 30 drawn from a normal distribution. The output is the lower and upper bounds rounded to two decimals.

❓ data_output

intermediate

Result of Sampling Distribution Visualization Data

After running this code to simulate sampling distribution of the mean, what is the mean of the sample means array?

SciPy

import numpy as np
np.random.seed(1)
sample_means = [np.mean(np.random.normal(100, 15, 50)) for _ in range(1000)]
mean_of_means = round(np.mean(sample_means), 2)
print(mean_of_means)

A 99.85

B 100.0

C 101.5

D 98.0

💡 Hint

The sample means should be close to the population mean but may vary slightly due to randomness.

The average of the 1000 sample means is close to the population mean of 100 but not exactly 100 due to sampling variability.

❓ visualization

advanced

Interpreting a Histogram of Sample Means

Which option best describes the shape of the histogram generated by this code showing sample means from repeated sampling?

SciPy

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(2)
sample_means = [np.mean(np.random.exponential(scale=1.0, size=40)) for _ in range(1000)]
plt.hist(sample_means, bins=30, color='skyblue', edgecolor='black')
plt.title('Histogram of Sample Means from Exponential Distribution')
plt.xlabel('Sample Mean')
plt.ylabel('Frequency')
plt.show()

A The histogram is heavily skewed right, reflecting the original exponential distribution.

B The histogram shows multiple peaks indicating a bimodal distribution.

C The histogram is uniform because sample means are evenly distributed.

D The histogram is symmetric and bell-shaped due to the Central Limit Theorem.

💡 Hint

Think about how the Central Limit Theorem affects the distribution of sample means.

Even though the original data is exponential (skewed), the distribution of sample means tends to be normal (bell-shaped) when sample size is large enough.

🧠 Conceptual

advanced

Why Use Probability Distributions in Statistics?

Which option best explains why statistics uses probability distributions to quantify uncertainty?

A Because probability distributions provide exact predictions of future events without error.

B Because probability distributions model the variability and randomness in data, allowing us to estimate uncertainty.

C Because probability distributions eliminate the need for data collection by assuming fixed values.

D Because probability distributions simplify data by ignoring variability and focusing on averages.

💡 Hint

Think about what uncertainty means in real life and how statistics handles it.

Probability distributions represent how data can vary and help quantify the uncertainty inherent in measurements and predictions.

🔧 Debug

expert

Identify the Error in Confidence Interval Calculation

What error does this code raise when trying to calculate a 99% confidence interval for a small sample?

SciPy

import numpy as np
from scipy import stats

sample = np.array([5, 7, 8, 6, 9])
mean = np.mean(sample)
sem = stats.sem(sample)
conf_int = stats.t.interval(0.99, len(sample)-1, loc=mean, scale=sem)
print(conf_int)

A IndexError: list index out of range

B TypeError: unsupported operand type(s) for +: 'int' and 'str'

C ValueError: degrees of freedom must be positive

D No error, outputs the confidence interval tuple

💡 Hint

Check the degrees of freedom parameter passed to the t.interval function.

The degrees of freedom should be sample size minus one, but the code uses the full sample size, causing a ValueError.