Stats Midterm (L2 - L12)

Question 1

What is the difference between sample and population?

Answer 1

The population refers to the entire group of individuals, items, or data points of interest in a specific study.

A sample is a subset of the population, representing a smaller group of individuals or data points selected from the population.

Question 2

Describe the principles of statistics.

Answer 2

Sample —> Empirical Distribution —> Theoretical Distribution —-> Population

Lecture 2, Slide 9

Question 3

What are the four key components of descriptive statistics?

Answer 3

1. Measures of central tendency.
2. Measures of Dispersion (variability)
3. Measures of distribution shape

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4. Frequency Distribution

Lecture 2: Slide 13

Question 4

Define hypothesis testing.

Answer 4

Hypothesis testing is a method used to evaluate two competing claims about a population based on sample data, with the goal of determining whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Question 5

With regard to hypothesis testing, what is the significance level (α)?

Answer 5

The significance level (α) is the probability of rejecting the null hypothesis when it is actually true (a false positive).

Extra Notes: A common value is 0.05, meaning there is a 5% risk of rejecting the null hypothesis when it is true.

Question 6

With regard to hypothesis testing, what is the p value?

Answer 6

The p-value in hypothesis testing is the probability of observing the data (or something more extreme) assuming that the null hypothesis H0 is true. It quantifies the strength of the evidence against the null hypothesis: the smaller the p-value, the stronger the evidence that the null hypothesis may be incorrect.

Question 7

What is the relationship between p and α when rejecting or failing to reject the null hypothesis?

Answer 7

p < α —> reject the null hypothesis
p > α —> do not reject the null hypothesis

Question 8

What is the difference between a histogram and a cumulative histogram?

Answer 8

Histogram: Shows the individual frequency for each bin.

Cumulative Histogram: Shows the running total (cumulative frequency) for each bin and those that came before it.

Question 9

True or False:
The mean is sensitive to outliers.

Answer 9

True.

Lecture 2 - Slide 16

Question 10

Describe how a histogram could be used to find the median.

Answer 10

A histogram can be used to estimate the median of a dataset by visually analyzing the distribution of data across the bins. The median is the value that separates the dataset into two equal halves, meaning 50% of the data points lie below it and 50% lie above it.

Question 11

True or False:
The standard deviation is a measure of dispersion.

Answer 11

True.

Lecture 2 - slide 24

Question 12

Define DOF with respect to statistics.

Answer 12

The DOF is the number of values in the final calculation of a statistic that are free to vary.

Lecture 2 - slide 27

Question 13

What are the two basic measures of general shape?

Answer 13

1. Skewness
2. Kurtosis

Lecture 2- slides 30 to 33

Question 14

A normal distribution has a kurtosis of ________.

Answer 14

3

Lecture 2- Slide 33

Question 15

Define the probability density function (PDF).

Answer 15

The PDF f(x) defines the probabiity that the variate f has the probability x.

Lecture 2 - slide 37

Question 16

Define the cumulative distribution function (CDF).

Answer 16

The CDF F(x) is the sum of a discrete PDF or the integral of a continuous one.

It tells you the probability that F takes on a value of less than x.

Lecture 2 -slide 37

Question 17

Define the interquartile range (IQR).

Answer 17

The interquartile range (IQR) is a measure of statistical dispersion that represents the range within which the central 50% of the data points lie. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1) of a dataset.

Question 18

Define a uniform distribution.

Answer 18

A uniform distribution is a type of probability distribution where all outcomes are equally likely within a specified range. In a uniform distribution, each value within the range has the same probability of occurring. The distribution is “flat” because no single outcome is favored over others.

Question 19

Define the binomial distribution.

Answer 19

The binomial distribution is a discrete probability distribution that describes the likelihood of obtaining a fixed number of successes in a specified number of independent trials of a binary experiment, where each trial has only two possible outcomes: success or failure. It is widely used in situations where the outcome of interest is a count of successes over multiple attempts.

Source:

Question 20

Describe a Poisson distribution.

Answer 20

The Poisson distribution expresses the probability of a given number of events occuring in a fixed interval (of space or time), if these events occur with a known constant mean rate and independantly of the time since the last event.

Lecture 3- slide 24

Question 21

True or False:
The rate must be constant for a Poisson distribution to hold.

Answer 21

True.

Lecture 3 - slide 24

Question 22

True or False
As the number of trials goes to infinity and the sucess probability p goes to zero, the binomial distribution approaches the Poisson distribution.

Answer 22

True.

Lecture 3 - slide 25

Question 23

What is the z-distribution?

Answer 23

When the standard normal distribution has a mean of zero and a standard deviation of one.

Lecture 3 - slide 31

Question 24

When is the logarithmic-normal distribution appropriate?

Answer 24

When the data have a lower limit (precip, frequency of earhtquakes etc.) in which the distribution has significant skewness.

Lecture 3 - slide 39

Question 25

Describe students T distribution.

Answer 25

The Student’s t-distribution (or simply the t-distribution) is a probability distribution used in statistics that is similar to the normal distribution but accounts for small sample sizes or when the population standard deviation is unknown. It plays a critical role in hypothesis testing and confidence intervals, particularly in situations where the sample size is small.

Question 26

What is the difference between the tails of the normal and the T-distribution?

Answer 26

The t-distribution is symmetrical and bell-shaped, similar to the normal (Gaussian) distribution. However, it has heavier tails, meaning it has more probability mass in the tails than the normal distribution. This reflects a higher likelihood of extreme values occurring, particularly in small samples.

Question 27

True or False:
As the dof increase, the T-distribution approaches the normal distribution.

Answer 27

True.

Question 28

The mean of the t-distribution is _____.

Answer 28

zero

Question 29

What is the difference between the one sample t-test and the two sample t-test?

Answer 29

One-sample t-test: Used to determine if the mean of a single sample is significantly different from a known or hypothesized population mean.

Two-sample t-test: Used to compare the means of two independent samples to see if they are significantly different from each other.

Question 30

Define a “confidence interval.”

Answer 30

A confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the true population parameter (such as the population mean) with a certain level of confidence.

It provides an estimate of the uncertainty or variability in a statistic, helping to indicate how well the sample data represent the entire population.

Question 31

The chi^2 test is mainly used for ______.

Answer 31

Statistical hypothesis testing.

Question 32

What is the F-distribution primarily used for?

Answer 32

The F-distribution is a continuous probability distribution that arises frequently in the context of analysis of variance (ANOVA), regression analysis, and hypothesis testing, especially when comparing variances of two different populations. It is primarily used to test whether the variances of two populations are equal.

Question 33

What is the purpose of the z-score?

Answer 33

The z-score is used to measure how many standard deviations an individual data point is away from the mean of the distribution. It helps to standardize different data points, allowing comparisons across different normal distributions.

Question 34

When calculating the confidence interval, when should the t-test be used? The z-test?

Answer 34

> The T-Test should be used when the population standard deviation is unkonwn and the sample size is small.

> The Z-test should be used when the population standard deviation is known or if the sample size is large.

Question 35

Define ANOVA.

Answer 35

ANOVA (Analysis of Variance) is a statistical method used to compare the means of three or more groups to determine whether there are statistically significant differences between them. It helps to assess whether the variation in a dataset is due to differences in group means or due to random chance.

Question 36

Describe spectral analysis.

Answer 36

An equivalent representation of a data trend using a sum of different periodic function.

Lecture 7 - slide 45

Question 37

What is a corelation coefficient?

Answer 37

The correlation coefficient measures the strength and direction of the linear relationship between two variables.

Question 38

What is Bivariate statistics?

Answer 38

Bivariate statistics is a branch of statistics that involves the analysis of two variables simultaneously to understand the relationship between them.

Question 39

Define “Deviation.”

Answer 39

The difference between a random variable and its expected value.

Lecture 9 - slide 6

Question 40

Define “Variance.”

Answer 40

Expected value of the square of the deviation of a random variable. In other words, the square of the standard deviation.

Lecture 9 - Slide 6

Question 41

Define “residuals.”

Answer 41

Data - model prediction.

Question 42

Generally describe both bootstrapping and and jackknifing.

Answer 42

Jackknifing and bootstrapping are both resampling techniques used in statistics to estimate the properties (e.g., standard error, bias, confidence intervals) of a sample statistic, particularly when the underlying distribution of the data is unknown or difficult to calculate analytically. While both methods are based on resampling, they differ in how the resampling is performed and in their applications.

Question 43

Describe how Jackknifing works.

Answer 43

Jackknife involves systematically leaving out one or more observations from the sample at a time to create different subsamples.
For a sample size of n, it creates n subsamples, where each subsample consists of n−1 data points, excluding a single observation at a time.

The statistic of interest (e.g., mean, variance) is recalculated for each subsample, and the results are used to estimate properties like the variance or bias of the original statistic.

Question 44

Describe how Bootstrapping works.

Answer 44

Bootstrapping involves drawing random samples with replacement from the original dataset to create multiple new datasets, called bootstrap samples.

For a sample size of n, you create many new samples (often 1,000 or more), each of size n, by randomly selecting data points from the original sample, allowing for repetition (i.e., the same data point can appear multiple times in a bootstrap sample). The statistic of interest is calculated for each bootstrap sample, and these estimates are used to derive the distribution of the statistic, as well as properties like the standard error, confidence intervals, and bias.

Question 45

Define the samping interval in a time series.

Answer 45

The time spacing between two samples.

Question 46

Define the sampling frequency between two samples in a time series.

Answer 46

1 over the sampling interval.
f = 1/dt

Question 47

True or False:
An integral transform is best thought of as a continuous analog to a dot product.

Answer 47

True.

Lecture 10 - slide 9

Question 48

A Fourier transform of a time series is anaogously a set of ___________________

Answer 48

Complex valued coefficients indexed by frequency.

Question 49

True or False:
Fourier transforms are linear operators.

Answer 49

True

Lecture 10 - slide 21

Question 50

Convolution is __________ in fourier space.

Answer 50

Multiplication.

Lecture 10 - slide 25

Question 51

True or False:
Periodic extension with the comb function leads to the Fourier series.

Answer 51

True

Lecture 10 - slide 48

Question 52

What is “band limited” mean?

Answer 52

A signal is said to be band limited if the amplitude of its spectrum goes to zero for all frequencies beyond some threshold called the cutoff frequency.

Lecture 11 - slide 3

Question 53

The nyquist frequency is calculated as f_N = ________

Answer 53

f_N = 1 / 2\Delta t