Chapter 7 - Testing One Sample Mean

What is the sampling distribution of the mean an example of?

A sampling distribution

What are the features of the sampling distribution of the mean?

1) Expected mean of the distribution is equal to the population mean 2) An approximate normal distribution (even if the distribution of scores in the population is not normally distributed) 3) Variability represented by standard error of the mean

What are the three factors that affect the decision about the null hypothesis?

1) Sample size 2) Alpha 3) Directionality of the alternative hypothesis

What are the two types of standard error of the mean?

1. Population standard error of the mean 2. Standard error of the mean

What is the variability of the sampling distribution of the mean represented by?

Standard error of the mean

What is central limit theorem?

A statistical principle that describes the modality, symmetry, and variability of the sampling distribution of the mean

If we want to test a study's research hypothesis, in order to understand the difference between the sample and what relationship exists in the population, what do we need to calculate?

An inferential statistic

What is a sufficiently large sample size defined as?

At least N=30

What is required in order to test a study's research hypothesis?

Calculating inferential statistics

What are the modality, symmetry, and variability of the sampling distribution of the mean defined by?

Central limit theorem

How do you make a decision about the null hypothesis after all calculations have been completed?

Compare the value of the z-statistic or t-statistic calculated with the critical values identified

What does using the Student t-distribution require calculating?

Degrees of freedom

What does a larger amount of variability of individual scores in the population mean for variability of sample means?

Larger amount of variability in sample means that are generated from this population

Which distribution does the z-test use?

Standard normal distribution

What does evaluating the difference between a sample mean and a population mean involve?

Determining the probability of obtaining a particular value of the sample mean

What do you need to indicate when stating alpha level?

Directionality of the alternative hypothesis

What is a sampling distribution?

Distribution of statistics for samples that are randomly drawn from populations

What is the sampling distribution of the mean?

Distribution of values of the sample mean for an infinite number of sample sizes N that are randomly selected from the population

What is the relationship between the directionality of the alternative hypothesis and the decision about the null hypothesis?

Greater likelihood of rejecting the null hypothesis when the alternative hypothesis is directional (one-tailed)

Why do we have to transform the sample mean into a particular statistic?

In order to determine the probability of getting the value of the sample mean in the population

Why is the word "error" used to describe variability (i.e., standard error of the mean) in the sampling distribution of the mean?

In theory, because all samples are drawn from the same population, the mean of every sample SHOULD be the same as the population mean. Any variability among the sample means is seen as the result of random factors or "error"

What do we mean by trying to determine the probability of getting our value of the sample mean under the assumption that the null hypothesis is true?

Null hypothesis being true = no effect exists We want to know if the probability of getting our sample mean is LOW enough to reject the null hypothesis (i.e., to falsify it!) and support that the effect exists and the difference between the sample mean an population mean is statistically signifcant

If the alternative is non-directional, and alpha is set at 0.05, is the region of rejection split into two halves or consolidated in one region?

Split into two regions, one at each tail of the distribution

What is calculated in order to describe the variability in a set of data?

Standard deviation

What is the difference between standard deviation and the standard error of the mean?

Standard deviation measures the variability of multiple individual scores, whereas standard error of the mean measures the variability of multiple sample means

Which distribution does the t-test use?

Student t-distribution

What is the purpose of testing one sample mean?

Test hypotheses regarding the difference between a sample mean and population mean

What does it mean for the sample means to be normally distributed?

That although there is variability among sample means, we expect the majority of them to be relatively close to the population mean (especially we decent sized samples)

What does an alpha level of 0.05 imply?

That the null hypothesis is REJECTED when the probability of getting the calculated value of a statistic is less than 0.05 (5%)

What is implied when the calculated statistic falls in the region of non-rejection?

That the statistic has a greater than alpha (5%) probability of occurring (i.e., it does NOT have a low probability of occurring)

What is implied when the calculated statistic falls in the region of rejection?

That the statistic has a low probability (less than 5%) of occurring

What is the standard error of the mean?

The average deviation of a sample mean from the population mean

What is testing one mean related to?

The difference between a sample mean and a population mean

What does the z-test for one mean test?

The difference between a sample mean and a population mean when the population standard deviation is known

What is the relationship between sample size and the decision about the null hypothesis?

The larger the sample size, the greater the likelihood of rejecting the null hypothesis

How does sample size affect sampling error?

The larger the samples used to calculate all the sample means, the smaller the effect of random, chance factors that create variability will be!

What is the relationship between alpha and the decision about the null hypothesis?

The larger the value of alpha, the greater the likelihood of rejecting the null hypothesis

What are degrees of freedom?

The number of values or quantities that are free to vary when a sample statistic is used to estimate a population parameter

What do we expect the mean of sample means to equal? Why?

The population mean because even though samples may differ from each other, we expect the mean of the sample means to be equal to the mean of the population.

For the sampling distribution of the mean, what do we expect the mean of the sample means to equal?

The population mean! Regardless of whether or not the samples differ from each other

What do the values of the z-statistic divide the distribution into?

The region of rejection and the region of non-rejection

What do we need in order to determine the probability of getting a particular sample mean value?

The sampling distribution of the mean

What is another way of interpreting the standard error of the mean?

The standard deviation (avg. deviation of a score from the mean) of the sampling distribution of the mean

What is the population standard error of the mean?

The standard deviation of the sampling distribution of the mean WHEN the population standard deviation for the variable is known

What is the standard deviation of the sampling distribution of the mean?

The standard error of the mean

Will there be more or less variability with smaller sample size?

There will be greater variability in smaller sizes because sampling error will have a stronger effect than on a larger sample

What is another way of interpreting the critical values?

They are values of the statistics that have our defined "low probability" or alpha

What do we use the probability of a statistic for?

To make a decision about the null hypothesis

Why is it important that the sampling distribution of the mean is approximately normal?

We can apply principles of the standard normal distribution to determine the probability of any particular score in the distribution

Why do we need to determine if the probability of a statistic is "low enough"?

We want to know if the probability of a statistic is low enough for us to decide that the difference between the sample mean and the population mean is statistically significant!

What does central limit theorem state?

When an infinite (endless) amount of random samples are drawn from a population, the sample means are approximately normally distributed with a mean equal to the population mean and a standard deviation equal to the standard error of the mean (assuming samples are of sufficient size)

Central limit theorem

When an infinite number of random samples are drawn from a population, the sample means are approximately normally distributed with a mean equal to the population mean and a standard deviation equal to standard error of the mean

When is the population standard error of the mean calculated?

When population standard deviation is known

When is the standard error of the mean calculated?

When population standard deviation is not known (and we have to estimate it from s)

When is the null hypothesis not rejected?

When the calculated statistic falls in the region of non-rejection

When is the null hypothesis rejected?

When the calculated statistic falls in the region of rejection

When is the z-test for one mean used?

When the population standard deviation is known

When is the t-test for one mean used?

When the population standard deviation is not known

Are the factors that affect the decision about the null hypothesis under the control of researchers?

Yes - to some degree

What are two statistical techniques used to test the difference between a sample mean and a population mean?

z-test for one mean t-test for one mean