Data Science Final

Question 1

SD of the sample mean

Answer 1

The SD of the sample mean measures how much, on avg, a sample mean is expected to vary from the true pop mean, i.e. how accurate our estimate of the mean is based on the sample size.

SD of sample mean = population SD / square root of sample size

Question 2

Which formula calculates the minimum sample size needed and/or the width of your confidence interval?

Answer 2

Question 3

What is z?

Answer 3

z is the number of SD’s away from the mean needed for your confidence level.

Question 4

An example of how confidence intervals and SD’s connect

Answer 4

Since 95% of all sample means fall within 2 SEs of the true mean, we can flip it around and say:
“There’s a 95% chance the true mean is within 2 SEs of my sample mean.”

Question 5

What is Chebyshev’s inequality?

Answer 5

Used for worst-case scenario no matter the data shape.

It’s more conservative than the normal curve and not used for precise confidence intervals.

Question 6

What is the relationship between confidence interval and p-value?

Answer 6

If the null value is outside the confidence interval, the p-value is smaller than the cutoff — so you reject the null.

But ^ only if the confidence level matches the p-value cutoff. If not, you can’t make a conclusion just from the interval.

Question 7

Matching confidence levels and p-value cutoffs

Answer 7

90% CI 10% cutoff (α = 0.10)
95% CI 5% cutoff (α = 0.05)
99% CI 1% cutoff (α = 0.01)

Question 8

Confidence Interval equation

Answer 8

ConfidenceInterval = SampleMean±(NumberofStandard Deviations × SDoftheSampleMean)

Question 9

What is the probability that ‘at least one’ of x occurs?

Answer 9

P(atleastone) = 1 − P(none)

Question 10

Correlation coefficient

Answer 10

r. Only measures linear association.

1. convert 2 variables to standard units
2. multiply
3. take the mean

Question 11

Slope of regression line

Answer 11

Only valid to calculate regression line when doing simple linear regression.

r * (SD of y / SD of x)

Question 12

Intercept of regression line

Answer 12

Only valid to calculate regression line when doing simple linear regression

avg. of y - slope * avg. of x

Question 13

estimate of y/fitted value

Answer 13

slope * x + intercept (in standard units)

Question 14

RMSE (Root Mean Squared Error)

Answer 14

1. square each error (actual y - predicted y)
2. take avg. (MSE)
3. take square root (so units match orig. values)

Question 15

minimize

Answer 15

The minimize function returns an array consisting of the slope & intercept that minimize the RMSE/MSE. We use this to calculate the regression line when we have a non-linear relationship or more than 1 x-variable.

Question 16

Residuals vs. Errors

Answer 16

Errors: true y (for pop) - predicted y. We cannot calculate this since true y is unknown

Residuals: actual y (from data) - predicted y (from regression line).

Question 17

Residuals plot

Answer 17

Plots the residuals against the predictor variable. The residual plot of a good regression shows no pattern - if there is a pattern, there may be a non-linear relationship between variables. This is because residuals and the predictor are uncorrelated.

Avg. of residuals is always 0; regression line is unbalanced

Question 18

Heteroscedasticity

Answer 18

Occurs when your regression predictions are less reliable for certain x-values. The spread/variance of residual changes as x increases (your errors are bigger or smaller depending on where you are on the x-axis).

Question 19

Signal line and noise

Answer 19

We assume there is a perfect straight-line relationship (the signal line) underneath all the randomness (noise).

Question 20

How do we know if our scatterplot/correlation from 1 random sample is representative of the true population?

Answer 20

1. Bootstrap the scatter plot as many times as the original sample size
2. Collect all slopes and draw an empirical histogram
3. Construct a 95% confidence interval

Question 21

How do we know if the slope of the true linear relation is 0?

Answer 21

If the interval contains 0, we cannot reject the null hypothesis that the slope of the linear relation is 0.

Question 22

Steps for Bootstrapping a Regression Confidence Interval

Answer 22

1. resample table using sample(). This creates a new bootstrap sample of the data
2. extract the x and y columns
3. compute the correlation between x and y
4. compute slope and intercept
5. use regression line to predict y for a given x
6. repeat above steps 10,000 times, storing predictions
7. if using 95% CI, take 2.5th and 97.5th percentiles

Question 23

Total Variation Difference (TVD)

Answer 23

Use when you’re comparing two categorical distributions, like bar charts of proportions (e.g., gender, color, vote choice). It measures how far apart the two distributions are — 0 means identical, 1 means completely different. TVD is used with permutation tests when the variable is categorical.

1. Calculate the proportion of each category in group 1 and group 2
2. Subtract those proportions for each category
3. Take the absolute value of each difference (so it’s all positive)
4. Add them up
5. Multiply by ½

Question 24

P-Value Formula

Answer 24

p-value for right-tailed test: P (teststatistic ≥ observed)

p-value for left-tailed test: 𝑃 (teststatistic ≤
observed)

p-value for two-sided test:
P (∣teststatistic∣≥∣observed∣)

Question 25

Overlaid Histograms

Answer 25

Use when: You want to compare distributions of a numerical variable across two or more categories (e.g., Prior Goals for wins vs. losses)

Question 26

Scatterplot

Answer 26

Use when: You're examining the relationship between two numerical variables

Question 27

Line Graph

Answer 27

Use when: You're tracking a numerical variable over time

Question 28

Bar Chart

Answer 28

Use when: You're showing the frequency or proportion of categorical values

Question 29

Steps of the bootstrap method for generating another random sample that resembles the population

Answer 29

treat the original sample as if it were the population draw from the sample, at random with replacement, as many times as the original sample size

Question 30

Left-tail distribution formula when bootstrapping confidence level

Answer 30

Question 31

right-tail distribution formula when bootstrapping confidence level

Answer 31

Question 32

kNN regression

Answer 32

Measuring the distance between a new data point and all old, but SAME, points in the training set (using Euclidean distance). Selecting the k closest (nearest) neighbors. Averaging the target values (e.g., prices) of those k neighbors to make a prediction.

Question 33

Euclidian distance

Answer 33