Chapter 12 - Data-Based and Statistical Reasoning

Question 1

Measures of central tendency:

Answer 1

measurements that describe the middle of a sample

Question 2

Outlier:

Answer 2

an extremely large or extremely small value compared to the other values

Question 3

Median:

1. what it is also known as
2. relationship to outliers
3. if mean and median are far from each other
4. if mean and median are very close
5. equation

Answer 3

1. Midpoint; where half of data points are greater than the value and half are smaller
2. Least susceptible to outliers, but not useful for data sets with very large ranges or multiple modes
3. If the mean and median are far from each other, implies the presence of outliers or a skewed distribution
4. If the mean and median are very close, this implies a symmetrical distribution

Question 4

Mode:

1. what it is
2. if a data set has two modes

Answer 4

1. Number that appears the most often in a set of data
2. If a data set has two modes with a small number of values between them, it may be useful to analyze these portions separately or to look for other variables that may be responsible for dividing the distribution into two parts

Question 5

Normal Distributions:

1. what is all the same
2. basis for what

Answer 5

1. All of the measures of central tendency are the same
   * We can transform any normal distribution to a standard distribution, with a mean of zero and a standard deviation of one*
2. Basis for the bell curve

Question 6

Skewed Distribution:

1. what they are
2. negatively skewed distribution

• where tail is
• mean and median relationship

3. positively skewed distribution

• where tail is
• mean and median relationship

Answer 6

1. Skewed distribution: one that contains a tail on one side or the other of the data set

2. Negatively skewed distribution

• Tail on the left (or negative) side
• Mean will be lower than the median

3. Positively skewed distribution

• Tail on the right (or positive) side
• Mean will be higher than the median

(in image: a = negative, b = positive)

Question 7

Bimodal Distributions:

Answer 7

Bimodal: a distribution containing two peaks with a valley in between

May only have one mode if one peak is slightly higher than the other

Question 8

Range:

1. what it is
2. does not consider what
3. relationship to outliers
4. relationship to standard deviation
5. equation

Answer 8

1. difference between its largest and smallest values
2. Does not consider the number of items of the data set
3. Heavily affected by the presence of outliers
4. Possible to approximate the standard deviation as one-fourth of the range
5. Range = xmax − xmin

Question 9

Interquartile range + Quartiles:

1. what they are
2. equation for IQR

Answer 9

Interquartile range: related to the median, first, and third quartiles

Quartiles: including the median (Q2), divide data into groups that comprise one-fourth of the entire set

2. The interquartile range is then calculated by subtracting the value of the first quartile from the value of the third quartile:

IQR = Q3 – Q1

Question 10

Standard Deviation:

1. can be used to determine what
2. what determines an outlier
3. on a normal distribution

• one standard deviation
• two standard deviations
• three standard deviations

Answer 10

1. Can be used to determine whether a data point is an outlier

2. If a data point falls more than three standard deviations from the mean, it is considered an outlier

3. On a normal distribution:

• 68% of data points fall within one standard deviation of the mean
• 95% fall within two standard deviations
• 99% fall within three standard deviations

Question 11

Reasons why outliers occur: (3)

Answer 11

1. A true statistical anomaly (ex: a person who is over seven feet tall)
2. A measurement errors (ex: reading the centimeter side of a tape measure instead of inches)
3. A distribution that is not approximated by the normal distribution (ex: a skewed distribution with a long tail)

Question 12

Independent events vs. Dependent events:

Answer 12

Independent events: have no effect on one another

ex: rolling a dice, picking it up, and rolling it again

Dependent events: do have an impact on one another, such that the order changes the probability

ex: container with five red balls and five blue balls, if you pick up one and don’t put it back, probability changes

Question 13

Mutually exclusive outcomes:

Answer 13

cannot occur at the same time

Ex: Cannot flip both heads and tails in one throw

Question 14

Exhaustive (when describing a group):

Answer 14

describes a group when there are no possible outcomes

Ex: flipping heads or tails are exhaustive outcomes of a coin flip; these are the only two possibilities

Question 15

Null hypothesis (H₀):

Answer 15

a general statement or default position that there is no relationship between two measured phenomena, or no association among groups

the hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error

Says that two populations are equal, or that a single population can be described by a parameter equal to a given value

Assumed to be true until evidence indicates otherwise

Question 16

Alternative Hypothesis:

1. Nondirectional
2. Directional

Answer 16

Alternative hypothesis: may be nondirectional or directional

Nondirectional: that the populations are not equal

Directional: ex - the mean of population A is greater than the mean of population B

Question 17

Test statistic:

1. what it is
2. what is also called

Answer 17

1. calculated and compared to a table to determine the likelihood that the statistic was obtained by random chance (under the assumption that our null hypothesis is true)

2. This is the p-value

Question 18

P-value is compared to what?

when it’s greater

when it’s less

Answer 18

a significance level (α); 0.05 is commonly used

If p-value is greater than α, then we fail to reject the null hypothesis

If p-value is less than α, then we reject the null hypothesis and state that there is a statistically significant difference between the two groups

Question 19

When the null hypothesis is rejected…

Answer 19

we state that our results are significantly significant

Question 20

Type I error & Type II error:

(Type II error - symbolized by what)

Answer 20

Type I error: likelihood that we report a difference between two populations when one does not actually exist

Type II error: occurs when we incorrectly fail to reject the null hypothesis

Likelihood that we report no difference between two populations when one actually exists

Symbolized by β

Question 21

Power:

Answer 21

the probability of correctly rejecting a false null hypothesis (reporting a difference between two populations when one actually exists)

Equal to 1 - β

Question 22

Confidence:

Answer 22

the probability of correctly failing to reject a true null hypothesis (reporting no difference between two populations when one does not exist)

Question 23

Confidence intervals:

Answer 23

reverse of hypothesis testing

We determine a range of values from the sample mean and standard deviation

We begin with a desired confidence level (95% is standard) and use a table to find its corresponding z or t score

Example: consider a population for which we wish to know the mean age. We draw a sample from that population and find that the mean of the sample is 30, with a standard deviation of 3. if we wish to have 95% confidence, the corresponding z-score (which would be provided on test day) Is 1.96.

• Thus the range is 30-3(1.96) to 30+(3)(1.96) = 24.12 to 35.88
• We can report that we are 95% confident that the mean age of the population from which this sample is drawn is between 24.12 and 35.88.

Question 24

Slope:

Answer 24

change in the y-direction divided by the change in the x-direction for any two points:

Question 25

Semilog graphs:

Answer 25

specialized representation of a logarithmic data set

They can be easier to interpret because the curved nature of the logarithmic data is made linear by a change in the axis ratio

One axis (usually x-axis) maintains the traditional unit spacing

Question 26

Correlation:

1. what it is
2. relationship to causation
3. if an experiment cannot be performed

Answer 26

1. refers to a connection - direct relationship, inverse relationship, or otherwise - between data
2. Correlation does not imply causation
3. If an experiment cannot be performed, we must rely on Hill’s criteria