Significance Testing,Sampling Distributions & Z-Scores

Question 1

What are descriptive stats

Answer 1

• they describe our data through measures of central tendency and measures of disperson.
• only tell us about our own data, cant generalise to population

Question 2

What do descriptive statistics ntot tell us

Answer 2

us whether the difference between these groups can be inferred beyond our sample to the population

Question 3

what do inferential statistics generate and what can they help with

Answer 3

p-value = help us understanding if there is a difference in our general population.

Question 4

what is a p-value/ inferential statistic dependant on

Answer 4

what type of data level is used

Question 5

What is an inferential statistic

Answer 5

• uses a random sample of data from pop to help make inferences about the pop.
  -helps make inferences that go beyond our data

Question 6

what are the two types of inferential statistics

Answer 6

frequentist ( focused on in this lecture)

bayes

Question 7

what is a null hypothesis

what is an alternative hypothesis

(BASIC UNDERSTANDING)

Answer 7

We define a null hypothesis. This means that there is no difference between the groups we are looking at.

An alternative hypothesis states that there is a difference in the results

Question 8

what do we compare the null hypothesis

Answer 8

with the alternative hypothesis to see a contrast

Question 9

What is the purpose of null hypothesis testing?

Answer 9

It estimates the probability of obtaining a result /pvalue (or one more extreme) by chance, assuming the null hypothesis of no difference or association is true.
If the result is unlikely or extreme under this assumption, we reject the null and conclude there is evidence of a real difference or association.

Question 10

for p-values what value is known as satistically significant

Answer 10

0.05/ 5%

Question 11

What is the alpha level

Answer 11

the leve whcih we accept a result to be significant (0.05 )

Question 12

what is the true definition of a p-value in frequent statistics

Answer 12

Probability the result we found (or one more extreme) occurred by chance assuming the null hypothesis is true.

Question 13

what is the misinterpretation about p-values

Answer 13

p-values tell us the likelihood that our (alternative) hypothesis is real / true. - It can’t be. The p-value is specific to an experiment and Null hypothesis.

Question 14

if p i slower than 5% what does this indicate

Answer 14

we should reject the null. As it is suprising enough to be real

Question 15

What does the normal distribution tell us in hypothesis testing?

Answer 15

Normal Distribution:

A bell curve that shows the range of possible outcomes.

Used to determine how likely your sample result is if the null hypothesis is true.

P-value:

The probability of obtaining a result as extreme (or more extreme) than your sample result, if the null hypothesis is true.

Low p-value (e.g., < 0.05) → Reject the null hypothesis (your result is unlikely under the null).

High p-value → Fail to reject the null hypothesis (your result is not unusual).

Null hypothesis (H₀): The average height is 160 cm.

Sample result: Average height = 165 cm.

P-value: 0.02 → Reject the null hypothesis (165 cm is unlikely if the true average is 160 cm).

Question 16

What percentages of observations fall within standard deviations of the mean in a normal distribution?

Answer 16

68% within ±1 SD

~95% within ±2 SDs

In a normal distribution, the ±1.96 standard deviations (SDs) from the mean cover 95% of the data, leaving only 5% outside of this range—split between the two tails of the distribution.

So in hypothesis testing, we often use 1.96 to determine the boundaries for a 95% confidence interval or the critical region for rejecting the null hypothesis at a 5% significance level (α = 0.05). This means that

Question 17

what are the region of rejections

Answer 17

They are the extreme ends (tails) of a distribution where results are unlikely if the null is true.
If a test statistic falls in this region (e.g., p < 0.05, beyond ±1.96 SDs), we reject the null hypothesis.

Question 18

What if a sample has an extreme (<5%) probability under the null hypothesis?

Answer 18

It’s very unlikely to happen by chance, so it might come from a different population — suggesting a real difference or effect.

Question 19

How do effect sizes (Cohen’s d) relate to p-values?

what is the definition of cohen d.

Answer 19

When Cohen’s d = 0 → no real effect → p-values are usually not significant (p > .05).

When Cohen’s d = 1–2 → large real effect → p-values are usually significant (p < .05).

Example: Height difference between men and women (d = 1.72) shows a strong, real difference that’s almost always statistically significant.

Cohen’s d is a measure of effect size that tells you how large the difference is between two groups (or conditions), relative to the variability within those groups.

Question 20

what is a 1-tailed test

Answer 20

used for directional hypothesis . The 5 percent significance levels is concentrated on one tail

Question 21

what is a 2-tailed test

Answer 21

non-directional hypothesis
5 percent significance levels are split across both tails. We only use 2-tailed tests are used

Question 22

what is the 5 percent significance zone

Answer 22

part of our results that are too extreme/ random

Question 23

What are Cohen’s d effect size bins

Answer 23

Cohen provided guidelines to help interpret the magnitude of the effect size, or what we often call effect size bins.

0.2 = small

0.5 = medium

0.8 = large

Question 24

How do effect size and p-value differ?

Answer 24

Effect size (d): how big the difference is.

p-value: how surprising the result is (depends on sample size).
✅ Big effects can be non-significant, and small effects can be significant.

Question 25

what are p-values dependant on

Answer 25

sample size- meaning that the power is small as we dont have enough data points,

Question 26

Describe the aim of sampling

Answer 26

to learn more about the population, the larger the sample , the more we learn about the population

Question 27

what is sample variability

Answer 27

Each time we draw a sample from the same population, we make different observations, meaning if we calculate a statistic for each sample. the statistic will also be different

Question 28

What happens if we take an infinite number of sample means?

Answer 28

we get a normal distribution of sample means because recording the average over random samples, means that the extreme values will be cancelled out leaving a normal distribution

Question 29

what is the sampling distribution of the mean

Answer 29

It’s the distribution of means from many samples of the same size drawn from a population. The mean of this distribution = expected value . Sample mean is equal to population mean The spread = standard error (SE)

Question 30

what des the standard error tell us

Answer 30

SE measures how much sample means vary from one sample to another due to random chance. Larger SE → more spread out sample means Smaller SE → sample means cluster closer to the population mean

Question 31

How does sample size affect the sampling distribution?

Answer 31

Small sample (e.g., 3): wider distribution, larger SE, noisier estimate Large sample (e.g., 15): narrower distribution, smaller SE, more accurate estimate

Question 32

What does the Central Limit Theorem (CLT) state

Answer 32

the means of many samples will alwaus form a normal distribution using the mean of the population and the standard error, even if the population isn’t normal.

Question 33

what are the implications of the central limit theorem

Answer 33

always has a normal distribution even if the underlying distribution is not , as well as being able to connect out sample scores to the population of values

Question 34

what is a thought experiment ( calculating standard error)

Answer 34

magining infinite samples to understand the sampling distribution.

Question 35

what does a thought experiment help with

Answer 35

connecting the sample with a population . every pop has a sample distribution of the mean for a specific sample sizes

Question 36

for calculating standard error what have mathematics found

Answer 36

the mean of the sampling distribution is the mean of population . the standard deviation of the sample is: d/ square root of n side ways d ( tail to right) - the sd of values in pop square root of n- the size of sample

Question 37

How do we estimate the standard error if we don’t know the population standard deviation?

Answer 37

We use the sample SD as an estimate. If the sample is large enough, this gives a good approximation for the standard error.

Question 38

what would the equation be

Answer 38

the standard deviation of the values in sample divided by the square root of the size of sample

Question 39

What is a 95% confidence interval?

Answer 39

An interval around the sample mean formed by adding and subtracting ±1.96 standard errors. It has a 95% probability of containing the population mean. ## Footnote range where u believe the true mean live

Question 40

the approximation for standard error can be used to calculate the condidence level of our sample . what does the interval tell us

Answer 40

It gives a range of plausible values for the population mean based on our sample. Shows how close the sample mean is likely to be to the population mean.

Question 41

What affects the width of a confidence interval and what does it show?

Answer 41

Wider CI if sample size is smaller (SE larger) Wider CI if population SD is larger Shows how close the sample mean is likely to be to the population mean

Question 42

What is a standard normal distribution (z distribution)?

Answer 42

A normal distribution that hasa been rescaled so it has with mean = 0 and standard deviation = 1.

Question 43

In a standard normal distribution, what interval contains 95% of values?

Answer 43

Approximately [-1.96, +1.96] standard deviations from the mean.

Question 44

Why is the standard normal distribution called a z distribution?

Answer 44

Because any normal distribution can be converted into a standard normal using a z-transformation, which rescales it to mean = 0 and SD = 1.

Question 45

how do we calculate a z transformation and why is it useful

Answer 45

Subtract the mean and divide by the SD for each value: z= value - mean /sd ​ Resulting z-scores tell how many SDs a value is from the mean. Allows comparison across different scales by normalizing the data.

Question 46

Why is the standard normal distribution useful?

Answer 46

By converting raw scores into standardized z-scores, we can use the standard normal distribution to make predictions and calculate probabilities for any normal dataset.

Question 47

what are another use of z-transofrmations

Answer 47

Another use of z transformation is to make scores from different scales comparable to infer things about behaviour