Exam 2 study Guide
A(n) ____________________ is based on our sample statistic; it conveys the range of sample statistics we could expect if we conducted repeated hypothesis tests using samples from the same population.
interval estimate
The sample with the _____ will contribute more to the calculation of the pooled variance estimate.
larger sample size
when two intervals
do not overlap, we conclude that the population means are likely different
a distribution of means is more tightly clustered
has a smaller standard deviation
Other things being equal, a larger sample size will _____ effect size.
have no effect on
calculate pooled variance We use all three degrees of freedom calculations, along with the variance estimates for each sample, to calculate pooled variance:
(Note: If we had exactly the same number of participants in each sample, this would be an unweighted average—that is, we could compute the average in the usual way by summing the two sample variances and dividing by 2.)
To calculate the standard deviation of the distribution of differences between means, we take the square root of the previous calculation, the variance of the distribution of differences between means. The formula is:
=√of mx+my
Which statement is true about final decisions make from a t test?
Confidence intervals provide more information than hypothesis tests.
What is the first step in calculating statistical power?
Determine the information needed to calculate statistical power- the population mean, the population standard deviation, the hypothesizes mean for the sample, the sample size and the standard error based on this sample size.
If a t score is calculated to be -.03 and our critical values are -2.365 and 2.365, what decision do we make about our research study?
Fail to reject the null hypothesis.
Other factors remaining constant, how does sM affect the width of a confidence interval?
If sM is large, then the confidence interval will be wider.
Other factors remaining constant, how does sdifference affect the width of a confidence interval?
If sdifference is large, then the confidence interval will be wider.
The formula for the degrees of freedom for the paired-samples t test is _____.
N - 1
The formula for the degrees of freedom for the paired-samples t test is _____.
N-1
null vs research hypothesis* what are they? Determine what they are in a given scenario.
Null = Boring -There will be no change or difference from an intervention Research = Exciting -For example, that a given intervention will lead to a change or a difference
Which of the following statements is a correctly worded null hypothesis for an independent sample t test?
On average, women categorize the same percentage of cartoons as funny as men do.
_______________ are inferential statistical analyses based on a set of assumptions about the population.
Parametric tests
_____ is a weighted average of the two estimates of variance that are calculated when conducting an independent-samples t test.
Pooled variance
Calculating a Confidence Interval for a Paired-Samples t Test
STEP 1: Draw a picture of a t distribution that includes the confidence interval. STEP 2: Indicate the bounds of the confidence interval on the drawing. STEP 4: Convert the critical t statistics back into raw mean differences. STEP 5: Verify that the confidence interval makes sense.
Standardization and the z-score
Standardization is a way to create meaningful comparisons between observations from different distributions. It can be done by transforming raw scores from different distributions into z scores, also known as standardized scores. A z score is the distance that a score is from the mean of its distribution in terms of standard deviations. We also can transform z scores to raw scores by reversing the formula for a z score. z scores correspond to known percentiles that communicate how an individual score compares with the larger distribution.
_____________________ is a measure of our ability to reject the null hypothesis given that the null hypothesis is false.
Statistical power
Paired-samples t-test difference single sample
The steps for the paired-samples t test are similar to those for the single-sample t test. The main difference is that for the paired-samples t test, we compare the sample mean difference between scores to the mean difference for the population according to the null hypothesis, rather than comparing the sample mean of individual scores to the population mean according to the null hypothesis, as we do when conducting a single-sample t test.
The numerator of the ratio for calculating all the t statistics contains:
a difference between means.
Confidence intervals for the independent-samples t test
are centered around the difference between means (rather than the means themselves)
According to your text book, all of the following are steps involved in calculating a single sample t-test EXCEPT: -identify the populations, distribution, and assumptions. Correct! calculate the z scores - state the null and research hypotheses -determine the critical values, or cutoffs
calculate the z scores
what is Cohen's d
d=(M-μ)/σ
calculated effect size?
d=[(Mx-My)-(μx-μy)]/s Pooled
A paired samples t test is also known as a(n):
dependent samples t test
The formula for degrees of freedom for a single sample t test is:
df = N - 1
As sample size decreases, the shape of the t distribution
gets progressively wider.
The only difference in the t statistic and the z statistic is:
in the denominator.
The difference between the denominator of the z test and that of the single-sample t test is that:
in the z test, we divide by the actual population standard error (σM), but in a t test, we divide by the estimated standard error ( s M)
In a between-groups research design with two groups, the appropriate hypothesis test is a(n) _____.
independent-samples t test
Single sample t-test
is a hypothesis test in which we compare a sample from which we collect data to a population for which we know the mean but not the standard deviation.
z-distribution
is a normal distribution of standardized scores—a distribution of z scores N(0,1)
With very few degrees of freedom, the test statistic:
needs to be more extreme to reject the null hypothesis
A(n) ___________________ is a summary statistic from a sample that is just one number as an estimate of the population parameter.
point Estimate
The standard deviation of a distribution of means is called the:
standard error.
z-test use μM=μ Standard Error σM=σ/√n Test Statistic z=(M-μM)/σM
We conduct a z test when we have one sample and we know both the mean and the standard deviation of the population. We must decide whether to use a one-tailed test, in which the hypothesis is directional, or a two-tailed test, in which the hypothesis is nondirectional. One-tailed tests are rare in the research literature.
For the same number of observations, degrees of freedom will always be _____ for an independent-samples t test as compared with a paired-samples t test.
twice as large
P Value if n is large, we can apply Central limit Central Limit theorem
use standard normal to get p value N(0,1) regardless of the form of the population distribution
P Value if n is Small, confident the populations distribution is Normal
use t distribution with n - 1 degrees of freedom to calculate p value t(n-1)
One difference between calculating an independent samples t test for a paired samples t test is that with the independent samples t test, we use the:
variance, not the standard deviation.
Hypothesis Testing Decisions: Reject the Null Hypothesis or fail to reject the Null Hypothesis -under what conditions do you do each of these?
we either reject the null hypothesis if the test statistic is beyond the cutoffs, or we fail to reject the null hypothesis if the test statistic is not beyond the cutoffs.
95% Confidence interval for μ
x̅ (+ or -) Cutoff*(s/√n), where cutoff comes from t v (N-1) small n assume its normal distribution large n no worry
A normal distribution of standardized scores is called the:
z distribution.
Transform z-score to raw score and vice versa
z=(X-μ)/σ X = z(σ) +μ.
The symbol representing a standard deviation calculated using a sample to estimate the population standard deviation is _____
σ
Calculate Cohen's d
1. σM=σ/√N 2. z=(M-μM)/σM 3. d=(M-μ)/σ
An independent samples t test is used with which type of research design?
A between groups design
How is the formula for Cohen's d different from the formula for the independent-samples t test?
In the denominator, we use the standard deviation for the difference between means rather than standard error
When calculating a confidence interval, you find that 0 occurs within your interval. What does this indicate?
It is plausible that the two distributions are identical.
The formula for standard deviation when estimating from a sample is:
Replace x bar with M We calculate estimated standard error by dividing by N − 1, rather than dividing by N, when calculating standard error.
Statistical power is the probability that we will reject the null hypothesis if we should reject it.
Statistical power is affected by several factors, but most directly by sample size.
Standardize a randome variable
Subtract off its mean then divide by its SD
Normal curve and percentiles
The Normal Curve and Percentages The standard shape of the normal curve allows us to know the approximate percentages under different parts of the curve. For example, about 34% of scores fall between the mean and a z score of 1.0. p(μ-σ<y<μ+σ)=0.68 p(μ-2σ<y<μ+2σ)=0.95 ; p(μ-3σ<y<μ+3σ)=0.997
The Normal Curve
The normal curve is a specific, mathematically defined curve that is bell-shaped, unimodal, and symmetric. The normal curve describes the distributions of many variables. As the size of a sample approaches the size of the population, the distribution resembles a normal curve (as long as the population is normally distributed).
z=(x-μ)/σ (z Statistic)
The symmetry of the z distribution makes it easy to calculate confidence intervals.
there are three degrees of freedom calculations for an independent-samples t test. We calculate the degrees of freedom for each sample by subtracting 1 from the number of participants in that sample:
dfX = N − 1 and dfY = N − 1. Finally, we sum the degrees of freedom from the two samples to calculate the total degrees of freedom: dftotal = dfX + dfY.
When using an independent samples t test we have to create a:
distribution of differences between means.
When conducting a study of gender differences, we have to employ a(n):
independent sample t test.
Cutoffs confidence interval using standard normal N(0,1) because of the CLT once we know the cutoff we can plug it into the general formula x̅ ± cutoff * (s/√n) or Estimate ± cutoff * (s.d of estimate)
replace the 1.96 (95% confidence Interval) with: the new % SD from Mean like 80% is within 1.28 SD of mean once we have the corresponding cutoff
to calculate the variance of the distribution of differences between means, we sum the variance versions of standard error that we calculated in the previous step:
s2difference = s2MX + s2MY = 0.211 + 0.169 = 0.380
A(n) ___________________ is a type of t test in which we compare data from one sample to a population for which we know the mean but not the standard deviation.
single-sample t test
Identify the formula for the paired-samples t test.
t=(Mdifference -0)/SM
The assertion that a distribution of sample means approaches a normal curve as sample size increases is called:
the central limit theorem.
When we know the population mean, but not the population standard deviation, which statistic do we use to compare a sample mean with the population mean?
t
When calculating a confidence interval for an independent-samples t test, what value should be at the center of the interval?
the difference between sample means
Which of the following is NOT needed in order to calculate the pooled variance for the independent-samples t test? the means for each group the total degrees of freedom the variance for each group the degrees of freedom for each group
the means for each group
degrees of freedom
the number of scores that are free to vary when we estimate a population parameter from a sample.
To understand statistical power, we need to consider several characteristics of the two populations of interest:
the population we believe the sample represents (population 1) and the population to which we're comparing the sample (population 2). We represent these two populations visually as two overlapping curves
When we cant use the CLT When n is Small
use the t distribution uses the Degrees of Freedom
Which of these is the correct formula for calculating effect size for a single sample t test?
Cohen's d=(M-μ)/s
How is a distribution of means different from a distribution of raw scores?
The distribution of means is more tightly packed
Assuming critical values of -2.306 and +2.306 for an independent-samples t test, if we obtain a calculated t value of -2.30, we:
-2.30 should fail to reject the null hypothesis
Hypothesis testing decisions using test statistics applies to all hypothesis testing
1) Assume one hypothesis is true and check it with a large sample size and a test statistic with which the Central Limit Theorem applies if Null is true the test Statistic will be in the middle part if the test statistic is in the tail then we may have made the wrong assumption. look at p-value if its small the null is probably not true thresh-hold .05
Five Factors That Affect Statistical Power Name three ways to increase statistical power. increase your alpha, turn a two-tailed hypothesis into a one-tailed hypothesis, and increase N
1) Increase in alpha we see how statistical power increases when we increase a p level of 0.05 to 0.10 This has the side effect of increasing the probability of a Type I error from 5% to 10%, however, so researchers rarely choose to increase statistical power in this manner. 2) Turn a two-tailed hypothesis into a one-tailed hypothesis. We have been using a simpler one-tailed test, which provides more statistical power. 3) Increase N the increase in sample size, makes means more normal which results in less overlap = more statistical power 4) Exaggerate the mean difference between levels of the independent variable. larger difference in means results in less overlap between curves, less overlap more statistical power 5) decrease standard deviation The curves can become narrower not just because the denominator of the standard error calculation is larger, but also because the numerator is smaller. When standard deviation is smaller, standard error is smaller, and the curves are narrower. We can reduce standard deviation in two ways: (1) by using reliable measures from the beginning of the study, thus reducing error, or (2) by sampling from a more homogeneous group in which participants' responses are more likely to be similar to begin with.
A researcher is conducting an independent samples t test. She calculates the s2MXas 1.1 and the s2MYas 0.95. Based on these values, what is the value of the standard deviation of the difference between means (sdifference)?
1.43 s2MXas (1.1) + s2MYas (0.95) =2.05 take Square root of 2.05=1.43
The results of an independent-samples t test yield the following information: MX = 42, MY = 36, spooled = 3.2. What is the calculated effect size?
1.88 d=[(Mx-My)-(μx-μy)]/s Pooled =[(42-36)-0]/3.2=1.875
Sample Size and Statistical Significance
A statistically significant result is not necessarily one with practical importance. As sample size increases, the test statistic becomes more extreme and it becomes easier to reject the null hypothesis. Effect size for a z test is measured with Cohen's d, which is calculated much like a z statistic, but using standard deviation instead of standard error. A meta-analysis is a study of studies that provides a more objective measure of an effect size than an individual study does.
what is a z-score
A z score is the number of standard deviations a particular score is from the mean. A z score is part of its own distribution, the z distribution, z scores give us the ability to convert any variable to a standard distribution, allowing us to make comparisons among variables.
Calculate Effect Size μM=μ σM=σ/√N z=(M-μM)/σM As sample size increases, so does the test statistic (if all else stays the same). Because of this, a small difference might not be statistically significant with a small sample but might be statistically significant with a large sample.
Because effect size is a standardized measure based on scores rather than means, we can compare the effect sizes of different studies with one another, even when the studies have different sample sizes Increasing sample size always increases the test statistic if all else stays the same. Notice that each time we increased the sample size, the standard error decreased and the test statistic increased.
For an independent samples t test, which statistic do we use to measure effect size?
Cohen's d
What measure of effect size assesses the difference between two means in terms of standard deviation?
Cohen's d
Degrees of freedom total
Df total = N-1
Standard Error σM=σ/√N The standard error is the standard deviation of the population divided by the square root of the sample size, N. The formula is:
Fortunately, there is a simple calculation that lets us know exactly how much smaller the standard error, σM, is than the standard deviation, σ. the larger the sample size, the narrower the distribution of means and the smaller the standard deviation of the distribution of means—the standard error. We calculate the standard error by taking into account the sample size used to calculate the means that make up the distribution.
Hypothesis testing sample of data & null hypothesis & alternative hypothesis we compute a test statistic then a p value based on the test statistic under the null hypothesis
H0: μ=x Ha: μ does not = x we can use a standardized x bar μ = 0 σ = 1
Using Standard Error to Calculate the t Statistic
The formula for the single-sample t statistic is:
When calculating a confidence interval for an independent-samples t test, you find that 0 occurs within your interval. What does this indicate?
It is plausible that the two distributions are identical.
Calculating Effect Size for a Paired-Samples t Test d=(M-μ)/s
It is the same formula as for the single-sample t statistic, except that the mean and standard deviation are for difference scores rather than individual scores.
A z score has a known mean and standard deviation. What are they?
Mean = 0, SD = 1
Statistical Power Calculation table 8-2 increase power add participants
STEP 1: Determine the information needed to calculate statistical power—the hypothesized mean for the sample; the sample size; the population mean; the population standard deviation; and the standard error based on this sample size. STEP 2: Determine a critical value in terms of the z distribution and the raw mean so that statistical power can be calculated. STEP 3: Calculate the statistical power—the percentage of the distribution of means for population 1 (the distribution centered around the hypothesized sample mean) that falls above the critical value. z=(P1-P2)/σ/√N x=μ+zσ
Calculating Confidence Intervals with z Distributions
STEP 1: Draw a picture of a distribution that will include the confidence interval. -We draw a normal curve that has the sample mean, at its center, instead of the population mean. STEP 2: Indicate the bounds of the confidence interval on the drawing. STEP 3: Determine the z statistics that fall at each line marking the middle 95%. z=(x-μ)/σ STEP 4: Turn the z statistics back into raw means. σM=σ/√N Mlower = −z(σM) + Msample Mupper = z(σM) + Msample STEP 5: Check that the confidence interval makes sense.
The Six Steps of the Paired-Samples t Test
STEP 1: Identify the populations, distribution, and assumptions. STEP 2: State the null and research hypotheses. STEP 3: Determine the characteristics of the comparison distribution STEP 4: Determine the critical values, or cutoffs. STEP 5: Calculate the test statistic. STEP 6: Make a decision.
The Six Steps of Single Sample t-Test
STEP 1: Identify the populations, distribution, and assumptions. STEP 2: State the null and research hypotheses. STEP 3: Determine the characteristics of the comparison distribution. STEP 4: Determine the critical values, or cutoffs. STEP 5: Calculate the test statistic. STEP 6: Make a decision After completing the hypothesis test Write the symbol for the test statistic (e.g., t). Write the degrees of freedom, in parentheses. Write an equal sign and then the value of the test statistic, typically to two decimal places. Write a comma and then indicate the p value by writing "p =" and then the actual value
A researcher is conducting an independent samples t test for a study with 12 people in group 1 and 14 people in group 2. Her calculation of the t statistic yields a value of -2.21. For a two-tailed test with an alpha level of 0.05, what is the appropriate conclusion?
She should reject the null hypothesis. Because -2.21 is within the critical limit for the two tail test of 0.05 which is t= -2.365 and 2.365 if it had been more than this we would have accepted the null hypothesis
Statistical Power
Statistical power is the probability that we will reject the null hypothesis if we should reject it. Ideally, a researcher only conducts a study when there is 80% statistical power; that is, at least 80% of the time, the researcher will correctly reject the null hypothesis. Statistical power is affected by several factors, but most directly by sample size. Before conducting a study, researchers often determine the number of participants they need in order to ensure statistical power of 0.80. To get the most complete story about the data, it is best to combine the results of hypothesis testing with information gained from confidence intervals, effect size, and power.
Confidence intervals A confidence interval is one kind of interval estimate and can be created around a sample mean using a z distribution. The confidence interval confirms the results of the hypothesis test while adding more detail.
The confidence interval is centered around the mean of the sample. A 95% confidence level is most commonly used, indicating the 95% that falls between the two tails (i.e., 100% − 5% = 95%). Note the terms used here: The confidence level is 95%, but the confidence interval is the range between the two values that surround the sample mean.
The formula for standard error when we estimate from a sample is:
The formula for the t statistic for a single-sample t test is the same as the formula for the z statistic for a distribution of means, except that we use estimated standard error in the denominator rather than the actual standard error for the population.
Sample size and T Distribution
The t distributions (note the plural) help us specify how confident we can be about research findings. We want to know whether we can generalize what we have learned about one sample to a larger population. The t test, based on the t distributions, tells us how confident we can be that the sample differs from the larger population. The t distributions are more versatile than the z distribution because we can use them when (a) we don't know the population standard deviation, and (b) we compare two samples. Figure 9-1 demonstrates that there are many t distributions—one for each possible sample size
What is the difference between the z and t tests?
The t test uses the estimated standard error while the z statistic uses the actual standard error of the population of means
We use t distributions when we do not know the population standard deviation and are comparing only two groups.
The two groups may be a sample and a population, or two samples as part of a within-groups design or a between-groups design.
how do you calculate a z Score (steps)
We calculate the difference between an individual score and the population mean, then divide by the population standard deviation. z=(X-μ)/σ
What are the three assumptions for hypothesis testing?
dependent variable is measured on an interval or ratio scale, participants are randomly selected, and population distribution is approximately norma
test statistic = x̅ =Sample Mean [ ( x̅ - μ0)/(s/√n) ]
each sample would give a different sample mean is x̅ more likely true if the null hypothesis is true or if the alternative is true consider the sampling distribution of the test statistic for large samples we can use the central limit theorem even when its NOT Normal 1) Sampling distribution (known or approximated) when the Null Hypothesis is true SD = σ/√n which we can estimate with s/√n when testing the H0 that a mean μ=μ0, use [ ( x̅ - μ0)/(s/√n) ] sampling distribution for a large sample size Central Limit theorem ~ N(0,1) Small sample size CLT dose not apply specify form of population distribution if n is small, the distribution of [ ( x̅ - μ0)/(s/√n) ] is t n-1 (degrees of freedom) if the population distribution is normal this is due to the use of s to estimate sigma and the t distribution only applies if the population distribution is normal
Normal curve and sample size
few scores = guess at normal curve 30 scores = resemble normal curve 140 scores= resembles more and more like a normal curve we would see in the population.
Other things being equal, a larger difference between sample means will:
have no effect on the width of the confidence interval.
Other things being equal, a larger difference between sample means will _____ effect size.
increase
Effect Size
indicates the size of a difference and is unaffected by sample size.
Meta-Analysis
is a study that involves the calculation of a mean effect size from the individual effect sizes of many studies Step 1: Select the topic of interest, and decide exactly how to proceed before beginning to track down studies. Step 2: Locate every study that has been conducted and meets the criteria. Step 3: Calculate an effect size, often Cohen's d, for every study. Step 4: Calculate statistics—ideally, summary statistics, a hypothesis test, a confidence interval, and a visual display of the effect sizes (Rosenthal, 1995).
