STA 215 Test 1

Population

The entire set of people or objects that you want to \n draw conclusions about

Statistical Inference

Using the results of a sample to draw conclusions about a population, along with a measure of how good those conclusions are

Simple Random Sample

sample taken in such a way that every possible group of n individuals has the same chance of being the sample

Describing shape of a graph

Symmetry, modes, outliers

Variable

The characteristic being observed

Variables are denoted by

Capital letters

Value

The category or number of the variable for that individual

Values are denoted by

Lowercase letters

Two types of variables are:

Categorical and numerical

Ordinal variables

Can be categorical or numerical

Distribution of a variable tells you

Possible values of the variable and how often they occur

Area Principle

The areas for values in the picture must be proportional to the percent of observations which take on that value in the population or sample

Two pictures drawn to represent categorical variables

Pie charts and bar graphs

Does a bar graph have a shape? (Y/N)

N

Difference between bar charts using frequencies/relative

frequencies/percents

Labelling of y-axis. Relative areas will remain the same

To find the angle at the tip of a wedge in a pie chart

Multiply relative frequency by 360 degrees

Area principle holds over what kind of variables

Categorical and numerical

Symmetric

Left and right sides are mirror images of each other

Skewed

If a long tail is present, graph is skewed in the direction of the tail

Mode

Peak of a distribution

When there is more than one mode in the distribution, it usually means

There is more than one subgroup in the population

Outlier

A data point that falls away from the others, outside of the overall pattern

First thing to do when you find an outlier

Determine if the outlier is actually a correct data value

Two most common pictures for numerical data

Stemplots and histograms

If needed, stems in stemplot can be split into

2 or 5

What is the difference between the shape of a histogram and the shape of a stemplot?

None, they have the same shape

For small data sets, you should use a (stemplot/histogram)

Stemplot

For large data sets, you should use a (stemplot/histogram)

Histogram

Advantages of stemplot over histogram

Part of the original data can be recovered and it is fast to rank data from smallest to largest

Advantages of histogram over stemplot

Can make interval widths anything you want

Timeplot

Shows how a variable evolves over time

Does a timeplot have shape? (Y/N)

N

When analyzing a timeplot, you must analyze

Trends, cycles, and departures

Trends

A general up or down movement

Cycles

A recurring up and down movement

Departures

Outliers from the regular pattern

When you find a departure:

Look for a reason for it

It is possible to change the way that any picture looks by changing:

The scale

Why do categorical graphs not have a shape but numerical graphs do?

There isn’t an order to the variables in categorical data, but the variables are ordered in numerical data

Average

Measure of the center of the data set

Median

Middle number in the data set when numbers are ranked from smallest to largest

First step in finding the median

Rank the numbers from smallest to largest

Sample median is denoted by

m

Sample mean is denoted by

x bar

Mean can be found by

Adding up numbers and dividing by n

p% trimmed mean can be found by

Trimming the smallest and largest p% of data from the data set

What is the difference between a parameter and a statistic?

A parameter is calculated from the population; a statistic is calculated from a sample

What does it mean for a statistic to be resistant to the effect of outliers?

Large changes in a few of the data points do not change the value of the statistic by very much

If there are outliers in the data set, the (mean/median) is better for calculating the data set

Median

If there are not outliers in the data set, the (mean/median) is better for calculating the data set

Mean

Why is the mean better for calculating the average of a data set with no outliers?

It uses all the data, not just the middle values

A parameter is (constant/variable)

Constant

A statistic is (constant/variable)

Variable

In a picture of a distribution, the mean is

The balance point

In a picture of the distribution, the median is

Where the area is split in half

If the distribution is roughly symmetric, the mean and median will be

Close to the same number

If the mean and the median are the same number, the distribution is

Cannot tell

How do you know if data is skewed or has outliers?

Draw a picture

The pth percentile of a data set

a number such that at least p% of the numbers in the data set are at or below it and at least (100 – p)% of the numbers in the data set are at or above it

The 50th percentile is the

Median

5 numbers in the five number summary

Min, Q1, median, Q3, max

The 5 number summary splits the data into parts that contain x amount of the data

One fourth

Box plot

A picture of the five number summary

Lower fence

Q1 - (1.5)(IQR)

Upper fence

Q3 + 1.5(IQR)

Can you tell the shape of a distribution from a boxplot? (Y/N)

N

Biggest use of boxplots

Side by side comparison of data sets

If the data set has outliers or is skewed, the (IQR/standard deviation) is used

IQR

Spread

The average distance of all points from the middle

Mean deviation is always

Zero

Why is sample standard deviation divided by n - 1?

Samples tend to be less spread out than the populations that they are drawn from

Sampling distribution

Distribution of a variable

Experiment

Observation of a random phenomenon to see what happens

Trial

Repetitions of an experiment

Sample space

Set of all possible outcomes for an experiment

Event

A set of outcomes in the sample space S that possesses some characteristic

Disjoint/mutually exclusive

Two events with no outcomes in common and therefore cannot happen at the same time

Complement of A

All elements in S that are not in A

Probability of an event

Its long-term relative frequency

Independence

Two events are independent if knowing that one event happened does not affect the probability of another event

Law of Large Numbers

The long-run relative frequency of repeated independent events gets closer to the true relative frequency as the number of trials increases

The probability of P(A) of any event A is between

0 and 1

If S is the sample space, P(S) =

1

P(A U B) =

P(A) + P(B) - P(AB)

P(AB) =

P(A)P(B|A)

P(A|B) =

P(AB) / P(B)

Discrete random variable

Takes on a finite or countably infinite number of values

Continuous random variable

Can take on any value in an interval

On a continuous curve, probability that P = something is

Zero

z-scores always have mean

Zero

z-scores always have standard deviation

One

What % of observations are within one standard deviation of the normal curve?

68

What % of observations are within two standard deviations of the normal curve?

95

What % of observations are within three standard deviations of the normal curve?

99\.7

How can you tell if a data set has a normal distribution?

Draw a picture

Sampling distribution of a statistic

Distribution of the numbers you get if you take every possible sample of size n from the population and evaluate the statistic for each one

What does it mean for a statistic to be unbiased?

The mean of its sampling distribution equals the parameter it’s estimating

Why is the sample range an underestimate?

A sample cannot be bigger than the range, but can be smaller. On average, it will be too small

Spread of sampling distribution tells you:

Margin of error

Decrease margin of error by:

Increasing sample size