Statistics - Exam 2 11/3

Chapter 6,7,8,9

Probability

A situation in which several different outcomes are possible.

Ex: Deck of cards

Requirements for random sample

1. Each individual in the population has an equal chance of being selected.

   1. The probability of any one individual being selected from a population with N individuals; 1/N

      1. Ex: randomly selecting names from the Naperville household phone directly, random tables, local Walmart’s

2. The probabilities must stay constant from 1 selection to the next if more than one individual is selected.

   1. Requires sampling with replacement

   2. Independent random sampling

      1. Ex: Picking a card.

Sampling Error

A naturally occurring discrepancy between a sample statistic and population parameter.

• these differences occur by chance

• sample statistics vary from sample to sample

Sampling distribution

A distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

Distribution of sample means

the collection of sample means of all possible random samples of a particular size (n) that can be obtained from a population.

• All possible samples

• this is a distribution of sample means, not a distribution of individual scores

• it’s an example of sampling distribution

Central Limit Theorem

describes the distribution of sample means for any population (mean, sd, shape)

The characteristics of a distribution of sample means are…

1. shape

2. mean

3. variability

What is the shape characteristic in the distribution of sample means?

The distribution will be very close to normal if either, the population from which the samples are selected is a normal distribution and the number of scores (n) in each sample is relatively large (n > 30)

What is the mean characteristic in the distribution of sample means?

Cut and paste; it’s exactly equal to the population mean.

What is the variability characteristic in the distribution of sample means?

the standard deviation of the distribution of sample means is called the standard error of M

Standard error of the mean

measures the standard amount of difference between a sample mean and the population mean.

it tells us how well a sample mean estimates its population mean

What factors determine the standard error of the mean?

1. The size of the sample

   1. A large sample is more accurate than a small sample

   2. In general, the larger the sample size, the smaller the error between the sample mean and population mean

2. The standard deviation of the population from which the sample is selected.

What is the relationship between sample size and the standard error of the mean?

As the sample size increases, the standard error decreases and as the sample size decreases the standard error increases because you’re gathering less data. So, your sample mean becomes less reliable as it will be sensitive to random outliers.

Hypothesis test

a statistical procedure that uses sample data to evaluate a hypothesis about a population.

Describe a research situation in which a hypothesis test using a z-score would be appropriate?

A hypothesis test using a z-score is appropriate when you are comparing a sample mean to a known population mean, and the population standard deviation is strictly known.

For example, a researcher wants to know if mild brain stimulation has an effect on mathematical skill. To test this, they have a sample of 40 participants undergo mild brain stimulation before taking a nationally standardized math test. So, the standardized math test has a known population mean and a known population standard deviation from millions of past test-takers. Also, the sample size (n = 40) is greater than 30, satisfying the Central Limit Theorem. In all, the researcher is comparing the average score of their treated sample against the known national population standard to see if the brain stimulation caused a significant difference.

Describe Step 1 of a Hypothesis Test

1. States the hypothesis, it’s to know if it’s a null hypothesis or an alternative hypothesis

Describe Step 2 of a Hypothesis Test

2. Sets the criteria for a decision (draw a picture). This step chooses the alpha value and identifies the critical region.

Describe Step 3 of a Hypothesis Test

3. Collects data and computes sample statistics; compares your sample mean with the null hypothesis by calculating a z-score.

Describe Step 4 of a Hypothesis Test

4. Make a decision; use your z-score to make a decision about the null; there are two possible decisions. It’s either to reject the null hypothesis or fail to reject the null hypothesis.

Null Hypothesis (Ho)

States that the treatment has NO effect (entire population)

Alternative Hypothesis (H1)

States that there IS an effect of the treatment (increasing, decreasing)

2-tailed test

NO direction of effect specified

1-tailed test

Direction of the effect IS specified

Type 1 Error

A researcher REJECTS a null hypothesis that is actually TRUE

• example: crying wolf; when the villagers believe the boy who cried wolf and ran to help even though there was no wolf present.

Type 2 Error

A researcher FAILS TO REJECT a null hypothesis that is really FALSE

• example: failing to announce the wolf when the wolf actually there.

Alpha (level of signficance)

the PROBABILITY values to define the very unlikely sample outcomes (means) if the null is true.

Critical Region

Composed of EXTREME values that are very unlikely if the null is true.

• if sample data falls in the critical region, the null is rejected.

Beta

“false negative”

Power

the probability that the test will reject the null hypothesis if the treatment really has an effect; the probability that the test will identify a treatment effect if one really exists (1-Beta)

What is the relationship between alpha level and type 1 error?

the probability a hypothesis test will lead to a type 1 error equals the alpha level; selecting a SMALL ALPHA LEVEL lowers the risk of making a type 1 error, but it makes it more difficult to reject the Ho.

What does it mean to say a result is statistically significant?

A result is statistically significant if it is very unlikely to occur if the null hypothesis is true.

What’s effect size and why is it important?

it’s a significant treatment effect and it’s important because it does not always indicate a substantial treatment effect, and so statistically significant results are often accompanied by the effect size.

Identify the assumptions for hypothesis tests with z-scores

1. random sampling; helps ensure our sample is representative of the population we want to be able to generalize our findings.

2. independent observations; the occurrence of one effect has no effect on the probability of the second event.

3. the value of the population standard deviation is unchanged by the treatment

4. normal sampling distribution; the unit normal table can only be used if the distribution of sample means is normal.

Identify the assumptions for hypothesis tests with z-scores (part 2)

• as the mean difference INCREASES, the z-score INCREASES

• as the variability INCREASES, the z-score DECREASES

• as the # of sample scores INCREASES, the score INCREASES

Identify the factors that affect power that we discussed in class.

• as effect size INCREASES, power INCREASES

• as the sample size INCREASES, power INCREASES

• as alpha levels DECREASE, power DECREASES

• 2 tail test from 1 tail test, power INCREASES

• 1 tail test from 2 tail test, power DECREASES

t statistic

used to test hypothesis’s about an unknown population mean when the value of the standard deviation is unknown.

So, when we aren’t given population standard deviation we use this.

Describe degrees of freedom in relation to the t statistic.

it’s related to our certainty that the sample data is representative of the population of interest

• like the greater the degree’s of freedom for a sample, the better the sample variance represents the population variance.

Describe the t distribution.

• the shape of the t-distribution changes with the degrees of freedom

  • as the df approaches infinity, the shape of the t-distribution approaches the shape of the normal z-score distribution

• the t-distribution is more variable than the normal distribution because in the t-statistic, the sample variance changes from one sample to the next—which is not true with a z-score

• for t-statistics we use the t-distribution table

Assumptions of the singe-sample t test:

1. Independent observations

   1. the occurrence of one event has no effect on the probability of the second event

2. the population sampled must be normal because with larger sample sizes this can be violated.

Identify and discuss the factors that influence the single-sample t statistic.

• as the size of the mean difference INCREASES, t-statistic INCREASES

• as the variability of the score INCREASES, t-statistic DECREASES

• as the # of scores in a sample INCREASES, t-statistic INCREASES