psych statistics EXAM 2

For a population with µ = 56 and σ = 8, what is the z-score corresponding to X = 40?

40-56/8 = -2

size of standard error of M is determined by

(1) size of the sample (the law of large numbers) (2) variability (i.e., σ2 or σ) of scores in the population from which the sample is selected

A class consists of 60 males and 40 females. A random sample of n = 10 students is selected. If the first four randomly selected students were all females, what is the probability that the fifth student will be a female?

40/100= .40

For a population wit µ = 40 and σ = 4, what is the z-score corresponding to X = 46?

46-40/4 = 1.5

Which of the following accurately describes a hypothesis test statistical method?

An inferential statistical method that uses the data from a sample to draw inferences about a population

What position in the distribution corresponds to a z-score of z = - 1.5?

Below the mean by a distance equal to 1.5 standard deviation

For an exam with a mean of M =80 and a standard deviation of s = 6, Mary has a score of X = 88, Bob's score corresponds to z = +1.25, and Sue's score is located above the mean by 10 points. If the students are placed in order from lowest score to highest score, what is the correct order?

Bob, Mary, Sue.

characteristics of the distribution of sample means

Described by the mathematical proposition known as the Central Limit Theorem -shape -mean -standard deviation

Which of the following accurately describes the proportions of scores in the tails of a normal distribution?

Proportions in both tails are positive (i.e., the value of proportion is always a positive number)

Assumptions for hypothesis tests with z-scores

Random sampling The normal shape of sampling distribution Independent observations Value of standard deviation is unchanged by the treatment

If a hypothesis test produces a z-score in the critical region, what decision should be made?

Reject the null hypothesis.

What is measured by the numerator of the z-test statistic?

The actual distance between a sample mean M and the population mean µ.

The statements of the null and alternative hypotheses refers to which of the following?

The population after treatment.

normal distribution

- shape is almost perfectly normal - the population from which the samples are selected - the size of each sample is large

requirement for a random sample

-Every individual has an equal chance of being selected. -The probabilities cannot change during a series of selections. -There must be sampling with replacement.

Type I Error (i.e., α error)

-Researcher rejects a null hypothesis that is actually true -Researcher concludes that a treatment has an effect when it has none -Alpha level (significance level) is the probability that a test might lead to a Type I error.

A random sample of n = 16 scores is obtained from a population with a mean of µ = 60 and a standard deviation of σ = 10. If the sample mean is M = 55, what is the z-score for the sample mean?

-calculate standard error 10/ square root of 16 = 2.5 -find z score 60-55/2.5 = 2 answer: z= -2

For a normal population with a mean of µ = 80 and a standard deviation of σ = 10, what is the probability of obtaining a random sample of n = 25 scores with the mean greater than M = 85?

-calculate standard error 10/square root of 25= 2 -find z score 80-85/2 = -2.5 -go to z=2.5 on stat table and go to column C answer: .0062 or .62%

A normal distribution has a mean of µ = 60 with σ = 10. What score separates the highest 10% of the distribution from the rest of the scores? (Note: use the closest value to the top 10% listed in the Unit Normal table).

-calculate z score for each X value given in the answer choices -go to stat table and look in column C and find which is closest to 10% answer X=72.8

A sample of n = 81 is randomly selected from a population with μ = 50 with σ = 18. On average, how much error (i.e., difference) is reasonable to expect between the sample mean and the population mean?

-find the standard error of 18/square root of 81 = 2 answer: 2 points

A normal distribution has a mean of µ = 50 with σ = 5. If one score is randomly selected from this distribution, what is the probability that the score will be less than X = 35?

-find the z-score with the given info z=-3 -go to stat table in column D of 3 =.4987 -subtract .5000 (is the probability of what is contained in the lower half of the normal distribution) from .4987 =.0013 or .13%

IQ scores form a normal distribution with µ = 100 and σ = 15. Individuals with IQs above 140 are classified in the genius category. What proportion of the population consists of geniuses?

-find z score z=2.67 -go to column C .0038

A normal distribution has μ = 40 and σ = 4. What is the probability of randomly selecting a score smaller than 45 from this distribution?

-find z score from given info z=1.25 -find 1.25 on stat table and go to column B -answer= .8944 or 89.44%

A normal distribution has μ = 40 and σ = 4. What is the probability of randomly selecting a score smaller than 35 from this distribution?

-find z score z=-1.25 -find 1.25 on stat table and go to column C answer= .1056 or 10.56%

On an exam with M = 70, you have a score of X = 80. Which value of the standard deviation would give you the highest position in the class distribution?

-plug each standard deviation given in the answer choices into the sample z score formula - after calculating each sd conclude which has the highest position - in this case s = 5

A population has µ= 20. What value of standard deviation would make a score X = 25 an extreme value, out in the positive tail of the distribution?

-plug in each standard deviation given in the answer choices and plug them into the z score - in this case sd = 2 because 25-20/2 = 2.5 which is an extreme value

A population has µ = 50. What value of standard deviation would make X = 45 a central (average) score in the population ?

-plug in each standard deviation given in the answer choices and plug them into the z score -in this case sd = 10 because 45-50/10 = -.5 which is a central score in the population

A normal distribution of scores in population has µ = 500 and σ = 60. The distribution of sample means for samples of n = 36 randomly selected from this population will have an expected value of M equal ______ and a standard error equal _______.

-the distribution of sample means will be normal (because n>30) -expected value of 500 -standard error of standard deviation divided by the square root of M 60/square root of 36= 10 answer: 500, 10

expected value of M

-the mean of the distribution of sample means is equal to the mean of the population of scores - Mean of the distribution of sample means is called the expected value of M -M is an unbiased statistic

If all scores in a population with µ = 100 and σ = 10 are transformed into z-scores, than the distribution of z-scores will have a mean of _____ and a standard deviation of _____.

0, 1

What proportion of a normal distribution is located in the tail below z = - 1.5?

0.0668 (stat table column C)

What is the probability of randomly selecting a z-score greater than z = 0.50 from a normal distribution?

0.3085 or 30.85% (look in column C of .50)

What proportion of a normal distribution is located between the mean and z = 1.30?

0.4032 (stat table column D)

What proportion of a normal distribution is located between the mean and z = - 1.75?

0.4599 (always positive)

What proportion of a normal distribution is located in the body, below z = 1.25?

0.8944 (look at stat table column B)

What is the probability of randomly selecting a z-score less than z = 1.55 from a normal distribution?

0.9394 or 93.94% (found on column B of 1.55)

For a population with µ = 10 and σ = 4, what is the z-score corresponding to X = 12?

12-10/4 = .5

A researcher administers a treatment to a sample of participants selected from a population with µ = 40. If the researcher obtains a sample mean of M = 48, which value of standard error, σM will result in rejecting the null hypothesis (assume p < 0.05, 2-tails test)?

4

A normal distribution of scores in population has a mean of µ = 150 with σ = 25. A. What is the probability of randomly selecting a score greater than X = 160 from this population? B. If a sample of n = 100 is randomly selected from this population, what is the probability that the sample mean will be greater than M = 160?

A. Compute the z-score that corresponds to X = 160: z = (X - µ)/ σ = (160 - 150)/25 = 0.40 In the Unit Normal table find that the proportion of scores above z = 0.40 (i.e., in the tail of distribution, column C) equals 0.3446. Therefore, the probability of randomly selecting a score greater than X = 160 is 34.46%. B. Compute standard error σM of the sampling distribution for the samples of n = 100 σM = σ/SqRoot of n = 25/sq.Root of 100 = 25/10 = 2.5 Compute the z-score that corresponds to M = 160: z = (M - µ)/ σM = (160 - 150)/2.5= 10/2.5 = 4.0 In the Unit Normal table find that the proportion above z = 4.0 (i.e., in the tail of distribution, column C) equals 0.00003. Therefore, the probability of randomly selecting a sample with the mean greater than M = 160 is 0.003%.

Factors affecting the power of the test

As the effect size increases, power also increases Larger sample sizes produce greater power Reducing the alpha level (making the test more stringent) reduces power Using a one-tailed test increases power

A two-tailed hypothesis test is being used to evaluate a treatment effect with α = .05. If the sample data produce a z-score of z = - 1.56, what is the correct statistical decision and conclusion?

Fail to reject the null hypothesis and conclude that the treatment has no effect.

central limit theorem

For any population with mean μ and standard deviation , the distribution of sample means for sample size n will: - approach a normal distribution as n approaches infinity -have a mean equal population μ (i.e., expected value of M) -have a standard deviation of σ/√n (i.e., standard error)

A researcher wanted to investigate an effect of listening to hip-hop music on reading comprehension. She randomly selected a sample of n = 36 college students. The sample of students completed a reading comprehension test while hip-hop music was played in the background. In the college population, the mean reading comprehension test score is μ = 40 and σ = 18. The sample of n = 36 students who took the test with hip-hop music in the background had the mean reading comprehension score of M = 32. A. Do the data indicate a significant effect of hip-hop music on reading comprehension? Use two-tailed test with α = .05 to answer this research question.

H0: µ = 40 (There is no significant difference in reading comprehension of individuals who take the test with vs. without hip-hop music playing in the background . Alternative wording: There is no significant effect of hip-hop music on reading comprehension). H1: µ ≠ 40 (There is a significant difference in reading comprehension of individuals who take the test with vs. without hip-hop music playing in the background . Alternative wording: There is a significant effect of hip-hop music on reading comprehension). The standard error is σM = 18/SqRoot of 36 = 18/6 = 3.0 z = (M - µ)/σM = (32 - 40)/3 = - 8/3 = -2.67 For α = .05, the critical region consists of z-scores below/above ±1.96; therefore, the outcome of the z-test is in the critical region. Conclusion: (The H0 is rejected). The experiment showed a significant negative effect of hip-hop music on reading comprehension, z = -2.67, p < .05, 2-tailed test, d = -0.44.(Alternative wording: Hip-hop music produced a significant decline of reading comprehension, z = -2.67, p < .05, 2-tailed test, d = -0.44.. B. (3 points) Cohen's d = (M - μ)/σ = (32-40)/18 = -8/18 = -0.44. The hip-hop music has a medium size effect on reading comprehension.

Under what circumstances will the distribution of sample means have a normal distribution shape?

If the population distribution has a normal shape or if the sample size is greater than 30

A sample is randomly selected from a normal population with μ = 50 and σ = 12. Which of the following samples would be considered extreme and unrepresentative for this population?

M = 56 and n = 36

Type II Error (i.e., β error)

Researcher fails to reject a null hypothesis that is really false Researcher concludes that there is no treatment effect when it has been Probability that a researcher has failed to detect a real treatment effect

Which of the following accurately describes outcomes (i.e., sample means) in the critical region (i.e.,the rejection area) of the distribution of sample means for the z-test?

Sample means with a very low probability if the null hypothesis is true.

Last week, Sarah had exams in English and in Biology. On the English exam, the mean was µ = 50 with σ = 5 and Sarah had a score of X = 55. On the Biology exam, the mean was µ=60 with σ = 10, and Sarah had a score of X = 70. In which class Sarah's performance was better based on this outcomes?

Sarah did equally well in both classes.

effect size : Cohen's d

Statistically significant effects (i.e. p < .05) are not always substantial Effect size measures the magnitude of a treatment effect It is computed & reported only if a significant effect is found (i.e., H0 is rejected) mean difference/standard deviation

There is a sample of scores with M = 50 and s = 5. If this sample is standardized in order to create a new sample with M = 100 and s = 15, what would be the new value for each of the following scores from the original sample? (X): 35, 40, 55, 70

Step 1:Each of the original scores has to be transformed into a z-score. z = (X - M)/s, therefore: z1 = (35 - 50)/5 = -15/5 = -3 z2 = (40- 50)/5 = -10/5 = -2 z3 = (55 - 50)/5 = 5/5 = 1 z4 = (70 - 50)/5 = 20/5 = 4 Step 2: New scores are computed by plugging in the z-score of each original X score and the new mean & new standard deviation to the formula X = M + z*s. X1 = 100 + (-3 *15) = 100 - 45 = 55 X2 = 100 + (-2 * 15) = 100 - 30 = 70 X3 = 100 + 1 * 15 = 100 +15 = 115 X4 = 100 + 4 * 15 = 100 + 60 = 160

distribution of sample means

The distribution of sample means is the collection of sample means for all the possible random samples of a particular size (n) that can be selected from a population.

Define the distribution of sample means and describe what is measured by the standard error of M.

The distribution of sample means is the set of sample means obtained from all possible random samples of a specific size (n) selected from a particular population. The standard error of M is the standard deviation for the distribution of sample means; it measures the standard (i.e., average) difference between a sample mean M and the population mean µ.

A researcher randomly selected a sample of n = 100 individuals from a population with µ = 75 and administer an experimental treatment to the sample. What the researcher should expected if the treatment has no effect?

The sample mean should be relatively close to 75 and should lead the researcher to fail to reject the null hypothesis.

Properties of z-scores distribution

The shape of the z-score distribution is the same as the original distribution The mean of the z-score distribution is always 0 The standard deviation of the z-score distribution is always 1

A distribution of exam scores is negatively skewed with M = 75 and s = 8. If these students' scores are transformed into z-scores, what is the shape of z-scores distribution?

The z-scores distribution is negatively skewed.

For a population with µ = 300 and σ = 20, what is the X value corresponding to z = 1.50?

X= 300 + 1.50 (20) =330

For a sample with M = 40 and s = 16, what is the X value corresponding to z = - 0.25?

X= 40 + (-0.25)(16) = 36

For a population with µ = 60 and σ = 8, what is the X value corresponding to z = - 2.00?

X= 60 + -2 (8) = 44

A sample of n = 20 scores has a mean of M = 25 and a standard deviation of s = 4. In this sample, what is the z-score corresponding to X = 30?

Z= 30-25/4 = 1.25

A sample of n = 25 individuals is selected from a population with μ = 60 and σ = 10, and a treatment is administered to the sample. After treatment, the sample mean is M = 64. What is the size of the treatment effect evaluated by Cohen's d for this sample?

cohen's d = mean difference/standard deviation 64-60/10 = .4

A class consists of 75 females and 50 males. If one student is randomly selected from the class, what is the probability of selecting a male?

probability = # of desired outcomes/ # of total outcomes 50/125= .40 or 40%

A researcher administers a treatment to a sample of participants selected from a population with µ = 100. If after the treatment the sample mean is M = 91, which combination of factors will result in rejecting the null hypothesis (assume p < 0.05, 2-tailed test)?

standard deviation of 18 random sample of 36 -plug numbers into standard error equation and then find z score - z score = -3 which is less than 0.05

What is measured by the denominator of the z-test statistic?

standard error (i.e., the average distance between a sample mean M and the population mean µ that is expected if H0 is true)

Explain what is indicated by the sign of a z-score and what by its numerical value.

the sign of a z-score tells whether the score is located above (+) or below (-) the mean. the value of z-score tells the distance (difference) between the scores and the mean and express this distance in unites of standards deviations. ex. z=3 means is located 3 standard deviations above the mean

In general, it is easier to reject a null hypothesis and conclude that the treatment produced a statistically significant effect when one uses a 1-tailed test instead of 2-tailed test.

true

When using a z-test for hypotheses testing, in what situation a research has the best chance to find a statistically significant effect of an experimental treatment?

when after the treatment, there is a large difference between sample and population means, and the standard error is small.

For a population with µ = 10 and σ = 2, what is the X value corresponding to z = 2?

x= 10 + 2 (2) = 14

APA (American Psychological Association) reporting style:

z = computed value, p level, 1-tailed or 2-tailed test, effect size

Which of the following z-scores represent the location closest to the mean?

z= -0.25

For a population with a standard deviation of 8, what is the z-score corresponding to a score that is 4 points above the mean?

z= 0.5

What z-score value separates the highest 15% of the scores in a normal distribution from the lower 85%? (Note: use the closest value to the top 15% listed in the Un Normal table)

z= 1.04 (.1492 found in column C)

Which of the following z-scores correspond to a score that is above the mean by 3 standard deviations?

z= 3.00

A researcher is conducting an experiment to evaluate the effectiveness of cognitive-behavioral therapy for eating disorder. The mean eating disorder score in clinical population of people suffering from the disorder is μ = 60. The results of the experiment will be examined using a two-tailed hypothesis test. Which of the following is the correct statement of the null hypothesis, H0?

μ = 60 (i.e., the treatment has no effect).

A researcher is conducting an experiment to evaluate the effectiveness of treatment for depression in a population that is known to have a mean depression score of μ = 50. The results will be examined using a two-tailed hypothesis test. Which of the following is the correct statement of the alternative hypothesis, H1?

μ ≠50 (i.e., the treatment has a significant effect).