3070 Exam 2

To construct the sampling distribution, you must

- (1) identify the mean of the sampling distribution - (2) compute the standard error of the mean - (3) distribute the possible sample means 3 SEM above and below the mean

Alpha level or level of significance (α) Type I error

- (α) ___________ ___________ is the probability of rejecting the null hypothesis when the null hypothesis is true (i.e., the probability that we make an error)

Example: Standard Error of the Mean problem

- A random sample of 100 college students (population) are asked to state the number of hours they spend studying during finals week. The mean is 20 hours/week and the standard deviation is 15 hours/week.

Four steps of NHST: (Null Hypothesis Significance Testing)

1. State the null and alternative hypothesis 2. Set the criteria for a decision 3. Compute the test statistic 4. Make a decision

Null Hypothesis Significance Testing (NHST)

Purpose: - to test claims or ideas about the parameters for a group or population Goal: - determine the likelihood that a population parameter, such as the mean, is likely to be true

Chapter 8

Introduction to Hypothesis Testing

Chapter 9

One sample t-test and Independent t-test

3. Compute the test statistic (Four steps of NHST:)

Test statistic - A standardized value that is calculated from sample data (e.g., z or t) - We will use the test statistic to determine whether to reject the null hypothesis - Compares your data with what is expected under the null - Used to calculate the significance (p-value) - The larger the value of the test statistic, the further a sample mean deviates from the population mean stated in null hypothesis

Standard Error of the Mean

- Sampling error is measured using the standard deviation of the sampling distribution of sample means Called Standard Error of the Mean

Standard Normal Transformations With Sampling Distributions

- A sampling distribution can be converted to a standard normal distribution (of the sample means) by applying the z-transformation 𝑧=(𝑀−𝜇_𝑀)/𝜎_𝑀

One-Sample t Test

- A statistical procedure used to test hypotheses concerning a single group mean in a population with an unknown variance

Two-Independent Sample t Test

- A statistical procedure used to test hypotheses concerning the difference between two population means, where the variance in one or both populations is unknown - Making comparisons between two independent groups

Effect Size

- Size of an effect on a population

Three possibilities for the alternative hypothesis:

- Actual value is less than the assumed value - Children in the U.S. watch an average of less than 3 hours of TV per week H1: μ < 3 - Actual value is greater than the assumed value - Children in the U.S. watch an average of more than 3 hours of TV per week H1: μ > 3 - Actual value is different from the assumed value - Children in the U.S. watch an average that is different from 3 hours of TV per w. H1: μ ≠ 3

Random Sampling

- All individuals in a population must have an equal chance of being selected - The probability of selecting each participant must be the same - Avoids biased samples - Allows us to generalize from sample to population

beta

- Allows us to determine the probability of making a Type II error - The ________ is the probability of making a Type II error; symbolized as B.

Unbiased estimator

- An ______________ ______________ is a sample statistic obtained from a randomly selected sample that equals the value of its respective population parameter, on average

Sample Mean is an Unbiased Estimator

- An unbiased estimator is a sample statistic obtained from a randomly selected sample that equals the value of its respective population parameter, on average

2. Normal distribution is theoretical (Characteristics of Normal Distribution)

- Behavioral data typically approximates a normal distribution

6. The standard deviation can equal any positive value (Characteristics of Normal Distribution)

- Data can vary (SD>0), or not vary (SD=0)

Locating Proportions Between Two Values

- Data indicative of chewing gum improving academic performance by mean scores on a standardized math test of 20 ± 9 points. Assuming these data are normally distributed, what is the probability that a student who chews gum will score between 11 and 29 points higher the second time he or she takes the test? Step 1 To transform raw scores 11 and 29 11-20/9 = -1.00 29-20 = 1.00 Step 2 Find the z score 1.00 in Appendix B, then look in Column B for the proportions between -1.00 and the mean: p = .3413 p= .3413 + .3413 = .6826

4. Make a decision (Four steps of NHST:)

- Decide whether to reject or retain the null hypothesis: - If the obtained value is unlikely when the null hypothesis is true, we can reject the null - Otherwise, we retain the null Two methods: 1. Can use critical value obtained from table and compare this to test statistic - if test statistic is greater or less than 2. Can also use the p-value for the test statistic and compare this to the alpha level you have set (0.05)

4. Normal distribution is symmetrical (Characteristics of Normal Distribution)

- Distribution of data above the mean is exactly the same as below the mean

Sampling Distributions- Sampling Distribution of the Mean

- Distribution of the values of the sample mean for all possible samples

2. Set the criteria for a decision (Four steps of NHST:)

- Done by stating the level of significance (α) - alpha level is the probability of rejecting the null hypothesis when the null hypothesis is true (i.e., the probability that we make an error) - Typically the level is set at 5% in behavioral research - If the null hypothesis is true, the probability of obtaining a sample mean that is consistent with the alternative hypothesis is only 5%

Central Limit Theorem

- For samples of size 30 or more, the sample mean is approximately normally distributed. The larger the sample size, the better the approximation.

One-Tailed Tests

- Greater power - If value stated in null hypothesis is false, this test will make it easier to detect and reject - Can be difficult to justify - Can lead to Type III error Key words: Greater than or Less than, higher or lower

3. Mean, median, and mode are all located at the 50th percentile (Characteristics of Normal Distribution)

- Half of the data (50%) in a normal distribution fall above the mean, median, and mode, and half (50%) fall below

Sample size

- Law of large numbers - Increasing the number of observations or sample size will decrease standard error - The smaller the standard error, the closer a distribution of sample means will be to the population mean

Two-Tailed Tests

- Less Power - More conservative - More difficult to reject null hypothesis even when there really is an effect - Eliminates possibility of Type III error Key words: difference, changing

5. The mean can equal any value (Characteristics of Normal Distribution)

- Mean can equal any number from positive to negative infinity

Sample Mean is an Unbiased Estimator

- Mean of the sampling distribution of the mean = the population mean

Null Hypothesis (H0)

- No difference, No effect - No relationship between group size and helping - Accept or Reject

1. State the null and alternative hypothesis (Four steps of NHST:)

- Null Hypothesis (H0) - statement about the population parameter (such as the mean) that is assumed to be true - Starting point to determine if null is likely to be true or not - Example: Children in the U.S. watch an average of 3 h of TV per week H(0): u= 3 - Alternative hypothesis (H1) - statement that contradicts the null - We think null is wrong, H1 allows us to state what we think is wrong - Example: Children in the U.S. watch more or less than 3 hours of TV

Be sure you can clearly state your hypotheses using both words and symbols

- Null: Children in the U.S. watch an average of 3 hours of TV per week H0: μ = 3 - Alternative: Children in the U.S. watch an average that is different from 3 hours of TV per week H1: μ ≠ 3

7. Total area under the curve is equal to 1.0 (Characteristics of Normal Distribution)

- Proportions of the area are used to determine the probabilities for normally distributed data

Central Limit Theorem

- Regardless of the distribution of scores in a population, the sampling distribution of sample means selected from that population will be approximately normally distributed - At least 95% of possible Ms one could select fall within 2 SDs of μ (Empirical Rule)

Sample Design

- Specific plan or protocol for how individuals will be selected or sampled from a population of interest - Must address two questions: - Do we replace each selection before the next draw? - Does the order of selecting participants matter?

Standard Error of the Mean

- Standard Error of the Mean (SEM) - Variance of the sampling distribution of sample means - where σ2 is the population variance and n is the sample size

To locate the proportion for z score

- Step 1: Transform a raw score (x) into a z score - Step 2: Locate the corresponding proportion for the z score in the unit normal table (Appendix B, page 655)

8. The tails of a normal distribution are asymptotic (Characteristics of Normal Distribution)

- Tails of the distribution are always approaching x-axis but never touch it - this allows for possibility of outliers in a data set

One-Sample z Test

- Test requires us to find the probability that the obtained sample mean would have occurred if the null hypothesis is true - To locate the probability of obtaining a sample mean in a sampling distribution, we must know... - The population mean - The standard error of the mean

APA in Focus: Reporting the Standard Error

- The Publication Manual of the American Psychological Association (2009) recommends any combination of three ways to report standard error - In text, in a figure, or in a table. - The mean is typically reported with the standard error

Type II error

- The _______________ is the mistake we make when we fail to reject the null hypothesis when, in reality, it should be rejected; considered an "error of blindness"

Type I error

- The ___________________ is the mistake we make when we decide to reject the null hypothesis when, in reality, we should not reject it; considered an "error of gullibility".

Degrees of Freedom (df)

- The df for a t distribution are equal to the df of the sample variance: n - 1. - As the sample size increases, sample variance more closely resembles population variance - Degrees of freedom of the sample variance increases, the shape of the t distribution changes - The probability of outcomes in the tails become less likely and the tails approach the x-axis faster as n increases

Central Limit Theorem

- The distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution

Sampling error

- The extent to which sample means selected from the same population differ from one another

Type II error

- The more participants we have, the less likely it is that we will make a _________________ because our study will have greater power. - Researcher has failed to detect a real treatment effect.

(1) Identify the mean (The Standard Error of the Mean)

- The sample mean is equal to the population mean, so the mean of this sampling distribution is 20

1. Normal distribution is mathematically defined (Characteristics of Normal Distribution)

- The shape of a normal distribution is specified by an equation relating each score (along x-axis) with each frequency (along y-axis)

(2) compute standard error (The Standard Error of the Mean)

- The standard error is the population standard deviation divided by the square root of the sample size

Alternative hypothesis (H1)

- There is a change, difference

Example of APA in Focus: Reporting the Standard Error

- Those participants randomly assigned to calculate their hourly wage showed no increase in happiness after the leisure period (M=3.47, SEM=1.12) compared to before the leisure period (M=3.56, SEM=0.99)

Standard Normal Transformations With Sampling Distributions

- To locate the proportion, and therefore probability of obtaining a sample mean: - Transform a sample mean (M) into a z score - Locate the corresponding proportion for the z score in the unit normal table

Use z transformation

- Use transformation to locate where a score would fall in the standard normal distribution - Once you know location, standard normal distribution rules to find the probability

One-Sample z Test

- Used to test hypotheses concerning the mean in a single population with known variance - Serves as a simple intro to the logic and steps of hypothesis testing - Not used very often in real world b/c we usually don't know the population variance

Factors That Affect Standard Error

- Variance of the population - If σ2 increases, SEM increases - If σ2 decreases, SEM decreases

Standard Error of the Mean

- Variance of the sampling distribution ≠ variance in the population

Type I error

- When we decide to reject our hypothesis when it was actually true, this is called a _________________. - Researcher concludes that a treatment has an effect when it has none

Standard Normal Transformations With Sampling Distributions

- Z-transformation is used to determine the likelihood of measuring a particular sample mean from a population with a given mean and variance

z Score

- __ ___________ is the value on the x-axis of a standard normal distribution. The numerical value specifies the distance or standard deviation of a value from the mean

Z-transformation

- __-___________________ is used to determine the likelihood of measuring a particular sample mean from a population with a given mean and variance

Independent samples:

- ________________ ______________ refers to the selection of participants such that different participants are observed one time in each sample or group

Standard Normal Distribution-

- a normal distribution with a mean equal to 0 and a standard deviation equal to 1 - Distributed in z score units along the x-axis

Standard Normal Transformation

-________________ ____________ _________________ is a formula used to convert any normal distribution to a standard normal distribution with a mean of 0 and standard deviation of 1

Characteristics of Normal Distribution

1. Normal distribution is mathematically defined 2. Normal distribution is theoretical 3. Mean, median, and mode are all located at the 50th percentile 4. Normal distribution is symmetrical 5. The mean can equal any value 6. The standard deviation can equal any positive value 7. Total area under the curve is equal to 1.0 8. The tails of a normal distribution are asymptotic

Three assumptions made about One sample t-test:

1. Normality - assume data in the population being sampled is normally distributed 2. Random Sampling - assume that the data were obtained using a random sampling procedure 3. Independence - assume that probabilities of each measured outcome in a study are independent

What is the probability that John scored 107 or higher on the test?

107-121/21 =.67 z score = 1- .2514 = .7486

What is the probability that John scored 107 or lower on the test?

107-121/21= .67 z score= .2514

What is the probability that John scored between 65 and 107 on the test?

65-121/21 = -2.67 z score .0038 107-121/21 = -.67 z score .2514 .2514 -.0038 = .2476 p = .2476

Locating Raw Scores- The unit normal table can be used to locate scores that fall within a given proportion or percentile

In the general healthy population, IQ scores are normally distributed with . Assuming these data are normally distributed, what is the minimum score needed to be in the top 10% of this distribution of intelligence scores in this distribution? To locate the score: Step 1: Locate a z score associated with a given proportion in the unit normal table Step 2: Transform the z score into a raw score (x) Step 1 The top 10% of scores is the same as p = .1000 toward the tail Look for p = .1000 in Column C The z score is z = 1.28 A z score equal to 1.28 is the cutoff for the top 10% of data Step 2 Determine which score, x, in the distribution shown corresponds to a z score equal to 1.28. Since z = 1.28, you can substitute this value into the z transformation formula: 1.28= x-100/15 (15)1.28= x-100/15(15) 19.2=x-100 A score equal to 119.2 on the IQ test is the cut-off for the lowest score in the top 10% of scores in this distribution

Directional Tests (Alternative hypothesis is stated as greater (>) or less than (<) null hypothesis)

Lower-tail test (H1<H0) H0: μ ≥ some value H1: μ < some value Upper-tail test (H1>H0) H0: μ ≤ some value H1: μ > some value

Directional, Upper-Tail z Test Example

Null Hypothesis H0: μ ≤ 12 With the reading program, mean improvement is at most 12 points in general population Alternative Hypothesis H1: μ > 12 With the reading program, mean improvement is greater than 12 points in general population p = .0505 -> z = 1.64 p = .0495 -> z = 1.65 Critical z = 1.645

Chapter 6

Probability and Normal Distribution

Chapter 7

Probability and Sampling Distribution

Normal Distribution

_The theoretical distribution with data that are symmetrically distributed around the mean, median, and mode -Scores closer to the mean are more probable, or likely, than scores further from the mean - Behavioral data that researchers measure often tend to approximate a normal distribution

Cohen's d

𝑑= 𝑀−𝜇/SD