Quantitative data part 2 (week 12)

Standard deviation

measures the dispersion or spread of a dataset relative to its mean. it is the square root of the variance

It gives you a clear number that summarizes how scattered the entire set of data is, specifically using the average (the mean) as the starting point.

• It answers the question: "On average, how far away from the center are all my data points?"

• Standard Deviation is the number that tells you the typical distance of any point in your data set from the average.

Normal distribution

is a data point is significantly above or below the mean or part of expected variation

What is the rule of standard deviation

68% of data lie within -1 or +1 of the mean

95% of data lie within -2 or +2 of the mean

99.7 of data lie within -3 or +3 of the mean

Variance

quantifies the degree of spread or dispersions in a set of values, tells us how much individual data points deviate from the mean

gives you a number to show how scattered the data is

measures distance between each data point and the mean

Inferential statistics, hypothesis testing

1. makes an assumption about a population

2. test that assumption using a sample

what conditions must be met to use a t test?

random sampling

independent observations

normal distribution

continuous data

homogeneity of variances

Normal distributions

1. z distrubtuion

2. t distribution

3. F distribution

4. Chi squared distribution

T and Z tests

determine whether the mean value between means are statistically different

z test we must know the population standard deviation

t test used with smaller sample sizes

One sample T test

compare the mean of a sample with the known reference value

directional or non directional

ex: compare 202 finsihing times with the data from last 20 years

Two tailed t test

is there a statistically significant difference between the mean value of the sample and the population

one tailed t test

is the mean value of the sample significantly larger or smaller than the mean value of the population

Null hypothesis two tailed

the sample mean is equal to the reference population

null hypothesis one tailed

the sample mean is equal to or greater than or less than the mean of the reference population

hypothesis two tailed

the sample mean varies significantly from the reference value

hypothesis one tailed

the mean value of the sample is larger or smaller than the mean of the reference population

Independent sample t test

compare the mean of two samples to see if there is a significant variation between groups

hypothesis: there is significant difference between the mean values of both groups

Paired Sample T-test

Compare identical subjects pre- and post- intervention (one sample; two times)

there is a difference between the mean of the pairs

right tailed t test

• You suspect the true value is higher than the assumed value.

• Alternative Hypothesis (Ha​): Uses the > (greater than) sign.

  • Example: Ha​:μ>100 (The mean score is greater than 100.)

• Rejection Region: The area where a test result would be considered extremely high, which is on the right side of the distribution curve. If your sample result falls in this extreme right "tail," you reject H0​

A battery manufacturer claims its batteries have an average lifespan of 400 hours. A consumer group believes the actual average lifespan is longer than 400 hours.

left tail t test

• Simple Idea: You suspect the true value is lower than the assumed value.

• Alternative Hypothesis (Ha​): Uses the < (less than) sign.

  • Example: Ha​:μ<100 (The mean time is less than 100 seconds.)

A fast-food restaurant claims the average customer wait time is 5 minutes. The manager wants to test if a new ordering system has decreased this average wait time.

two tailed t test

• You suspect the true value is simply different from the assumed value, but you don't know (or care) if it's higher or lower.

• Alternative Hypothesis (Ha​): Uses the = (not equal to) sign.

  • Example: Ha​:μ=100 (The mean temperature is different from 100∘C

The sample mean varies significantly from the reference

value

how to calculate one sample t test

n (total population) -1

how to calculate sample t test

n 1 + n2 -2