Lecture 8 - Null hypothesis significance testing (A way to test the reliability of results)

Question 1

What is the coefficient of determination and how is it calculated?

Answer 1

The coefficient of determination (r^2) is the proportion of variance in the dependent variable (Y) that is predictable from the independent variable (X). It is calculated by squaring the correlation coefficient r. For example, if r = 0.62, the r^2 = 0.38

Question 2

Explain how the coefficient of determination is used to understand shared variance.

Answer 2

the coefficient of determination indicates the percentage of variance in one variable that can be explained by the variance in another variable. For instance, if r^2 = 0.30, the 30% of the variance in the dependent variable can be explained by the independent variable

Question 3

what is the sampling distribution of the mean, and what are its properties?

Answer 3

the sampling distribution of the mean is the distribution of sample means over repeated sampling from the same population. Its properties include:
- the mean of the sampling distribution (standard error, σM) depends on the population standard deviation (σ) and the sample size (N) and is calculated as:

Question 4

calculate the standard error for a population with σ=10 and a sample size of N = 25

Answer 4

see photo

Question 5

Given a population mean (μ) of 70 and a standard deviation (σ) of 20, what is the probability of obtaining a sample mean of 80 or higher from a sample of 25 people?

Answer 5

1. calculate the standard error: see photo
2. convert the sample mean to a z-score: see photo
3. Use a z-table to find the probability:
   The area above a z-score of 2.5 is 0.62%. Thus, the probability of obtaining a sample mean of 80 or higher is 0.62%

Question 6

within what limits would the central 95% of sample means fall for the same distribution and sample size?

Answer 6

• the central 95% of a normal distribution corresponds to z-scores of -1.96 and +1.96
• convert these z-scores to raw scores:
  
  • lower limit = 70 - (1.96 x 4) = 62.16
  • upper limit = 70 + (1.96 x 4) = 77.84
    Therefore, 95% of sample means would fall between 62.16 and 77.84

Question 7

What is the null hypothesis significance testing (NHST)?

Answer 7

NHST is a statistical method used to determine if there is enough evidence to reject a null hypothesis, which assumes that any observed difference is due to sampling error. The alternative hypothesis assumes that the observed difference is real.

Question 8

Explain the four possible decision outcomes in NHST

Answer 8

The four possible decision outcomes in NHST are:
- True Positive (Correct Rejection): The null hypothesis is false, and we correctly reject it/
- False Positive (Type I Error): The null hypothesis is true, but we incorrectly reject it
- True Negative (correct retention): The null hypothesis is true, and we correctly retain it
- False Negative (Type II Error): The null hypothesis is false, but we incorrectly retain it.

Question 9

What is the conventional threshold for statistical significance in NHST, and what does it mean?

Answer 9

The conventional threshold for statistical significance in NHST is 5% (p < 0.05). It means that is the probability of obtaining the observed result under the null hypothesis is less than 5%, we reject the null hypothesis and consider the result statistically significant

Question 10

Given a population IQ mean (μ) of 100 and standard deviation (σ) of 15, and a sample mean (M) of 110 from 4 students, assess the probability of obtaining this sample mean under the null hypothesis

Answer 10

1. Calculate the standard error: see photo
2. Convert the sample mean to a z-score: see photo
3. Use a z-table to find the probability:
   the probability of a z-score of 1.33 or more is approximately 9%. Therefore, the probability of obtaining a sample mean of 110 or higher due to sampling error is 9%, which is not statistically significant (p >0.05)