# Variables Distribution in Random Group – Statistics

### Variables Distribution in Random Group, Scale of Measurement – Statistics

The key terms in these Statistics chapters include Variables, Normal Distribution, Population, Correlation, Standard Deviation, Population Distribution, Regression Equation, Scale of Measurement, Sample, Random Group, Distribution. – Statistics

Identify the 2 variables in the following hypotheses, and suggest a way you could specifically measure each of them. Does exercise make people healthier?

The two variables in this hypothesis are “amount of exercise” and “healthiness.” You can measure this by taking a random group of people and having them increase their exercise by 30 minutes each day for a few weeks and track their results (lost weight, gained muscle).

What’s the scale of measurement? religious affiliation

Nominal

Delores must grade a driver’s test completed by a group of training students. She knows that driver’s test scores are normally distributed with µ = 50 and σ = 6. Students are required to score in the top 75% of students on the test in order to pass the test and continue on with the training course. Which score do students need to obtain at a minimum in order to pass the test and continue with the course?

X = 46

A highly selective university requires that applying students complete an entrance exam with µ = 60 and σ = 16. Applying students must avoid scoring in the bottom 89.5% of scores on the exam in order to be accepted into the university. Applicants must score more than _____ in order to be accepted to the university.

##### X = 80

A set of X and Y scores has M X = 4, SS X = 10, M Y = 5, SS Y = 40, and SP = 20. Which is the regression equation for predicting Y from X?

Ŷ = 2X – 3

A set of n = 15 pairs of scores ( X and Y values) produces a correlation of r = 0.40. If each of the X values are multiplied by 2 and the correlation is computed for the new scores, which value will be obtained for the new correlation?

#### r = 0.40

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

Proportions in both the left-hand and right-hand tails tend to be relatively small.

Last week, Sarah had exams in her math and Spanish classes. On the math exam, the mean was µ = 30 with σ = 5, and Sarah had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 8, and Sarah had a score of X = 68. For which class should Sarah expect the better grade relative to her peers in each class?

Math

A population has µ = 50. Which value of σ would make X = 55 an extreme value out in the tail of the population distribution?

#### σ = 1

For a sample of n = 20 scores, X = 25 corresponds to z = -1.20 and X = 40 corresponds to z = +1.80. What are the values for the sample mean and standard deviation?

M = 31 and s = 5

Identify the 2 variables in the following hypotheses, and suggest a way you could specifically measure each of them.
Does exercise make people healthier?

The two variables in this hypothesis are “amount of exercise” and “healthiness.” You can measure this by taking a random group of people and having them increase their exercise by 30 minutes each day for a few weeks and track their results (lost weight, gained muscle).

What’s the scale of measurement? religious affiliation

Nominal

Identify the 2 variables in the following hypotheses, and suggest a way you could specifically measure each of them. Does exercise make people healthier?

The two variables in this hypothesis are “amount of exercise” and “healthiness.” You can measure this by taking a random group of people and having them increase their exercise by 30 minutes each day for a few weeks and track their results (lost weight, gained muscle).

Delores must grade a driver’s test completed by a group of training students. She knows that driver’s test scores are normally distributed with µ = 50 and σ = 6. Students are required to score in the top 75% of students on the test in order to pass the test and continue on with the training course. Which score do students need to obtain at a minimum in order to pass the test and continue with the course?

##### X = 46

A highly selective university requires that applying students complete an entrance exam with µ = 60 and σ = 16. Applying students must avoid scoring in the bottom 89.5% of scores on the exam in order to be accepted into the university. Applicants must score more than _____ in order to be accepted to the university.

X = 80

A set of X and Y scores has M X = 4, SS X = 10, M Y = 5, SS Y = 40, and SP = 20. Which is the regression equation for predicting Y from X?

Ŷ = 2X – 3

A population is normally distributed with µ = 145 and σ = 20. Which is the percentile rank for X = 171?

90.32% (body)

A researcher tests their new treatment for sleep disorder among an individual that has a sleep disorder. The researcher knows that individuals diagnosed with sleep disorder tend to have sleep quality scores that form a normal distribution with µ = 40 and σ = 6. The researcher finds that this individual has a sleep quality score of X = 53 following treatment. Which is the probability that a score this large or larger would be obtained if the new treatment had no effect?

p = 0.0150 (tail)

A normal distribution has a mean of µ = 40 with σ = 10. If a vertical line is drawn through the distribution at X = 34, which proportion of the scores are on the left side of the line?

0.2743

What does SS stand for (in the context of statistics)?

Sum of squared deviations from the mean

Your sample data set has 4 scores in it. You calculate SS = 28. What is the variance?

#### 7

The standard deviation (SD) is most commonly used to get a sense of how far the typical score of a distribution differs from the mean. In computing the SD, why is it necessary to square the deviations from the mean for each score?

The mean of the unsquared deviations balances out to zero due to negative and positive values.

The ________ is equal to the square root of the ________.

Standard deviation; variance

Which of the following is true about an operational definition?

It is not the same as the construct itself

Dr. H. I. Strung has created a new way to measure anxiety and it turns out that people who score high on Dr. Strung’s measure show more anxious behavior than people who score low on the measure. What’s it called when this happens?

Predictive validity

One kind of validity is all about showing that we’re really measuring just one construct, rather than two constructs combined. Which one is that?

Divergent validity

Dr. N. Sight has developed a new intelligence test for school children. But she’s worried the test might be measuring reading ability rather than intelligence. In order to demonstrate that the test is really measuring intelligence and not reading ability, Dr. Sight demonstrate ____validity.

Divergent

An empirical journal article is an example of a ____ source.

Primary

All journal articles are primary sources.

#### False

What is the most relevant question you should ask when reading a results section?

Are appropriate statistics used?

HOMER

Hypothesize, operationalize, measure, evaluate, revise, replicate, report

Identify the 2 variables in the following hypotheses, and suggest a way you could specifically measure each of them. Does exercise make people healthier?

The two variables in this hypothesis are “amount of exercise” and “healthiness.” You can measure this by taking a random group of people and having them increase their exercise by 30 minutes each day for a few weeks and track their results (lost weight, gained muscle).

Which correlation should be used to measure the relationship between gender and grade point average for a group of college students?

Point-biserial correlation (used when one variable is dichotomous)

A researcher assesses self-esteem among college students at the beginning of the semester using a self-report questionnaire, as well as three months later at the end of the semester among the same sample of college students using the exact same self-report measure of self-esteem. The researcher determines that there is a significant positive relationship between self-esteem measured at the beginning and end of the college semester for the students sampled. This demonstrates how correlations can be used to demonstrate _____.

Reliability

As the sample size gets larger, the size of the correlation needed for significance _____

Gets smaller

An analysis of regression is used to test the significance of a linear regression equation based on a sample of n = 20 individuals. What are the df values for the F-ratio?

#### 1,18

A linear regression equation has b = 3 and a = – 6. What is the predicted value of Y for X = 4?

Ŷ = 6

Identify the 2 variables in the following hypotheses, and suggest a way you could specifically measure each of them.
Does exercise make people healthier?

The two variables in this hypothesis are “amount of exercise” and “healthiness.” You can measure this by taking a random group of people and having them increase their exercise by 30 minutes each day for a few weeks and track their results (lost weight, gained muscle).

What’s the scale of measurement? religious affiliation

Nominal

Delores must grade a driver’s test completed by a group of training students. She knows that driver’s test scores are normally distributed with µ = 50 and σ = 6. Students are required to score in the top 75% of students on the test in order to pass the test and continue on with the training course. Which score do students need to obtain at a minimum in order to pass the test and continue with the course?

X = 46

A highly selective university requires that applying students complete an entrance exam with µ = 60 and σ = 16. Applying students must avoid scoring in the bottom 89.5% of scores on the exam in order to be accepted into the university. Applicants must score more than _____ in order to be accepted to the university.

X = 80

A set of X and Y scores has M X = 4, SS X = 10, M Y = 5, SS Y = 40, and SP = 20. Which is the regression equation for predicting Y from X?

#### Ŷ = 2X – 3

The ________ is equal to the square root of the ________.

Standard deviation; variance

Which of the following is true about an operational definition?

It is not the same as the construct itself

Dr. H. I. Strung has created a new way to measure anxiety and it turns out that people who score high on Dr. Strung’s measure show more anxious behavior than people who score low on the measure. What’s it called when this happens?

Predictive validity

One kind of validity is all about showing that we’re really measuring just one construct, rather than two constructs combined. Which one is that?

Divergent validity

Dr. N. Sight has developed a new intelligence test for school children. But she’s worried the test might be measuring reading ability rather than intelligence. In order to demonstrate that the test is really measuring intelligence and not reading ability, Dr. Sight demonstrate ____validity.

Divergent

An empirical journal article is an example of a ____ source.

Primary

A set of n = 15 pairs of scores ( X and Y values) produces a correlation of r = 0.40. If each of the X values are multiplied by 2 and the correlation is computed for the new scores, which value will be obtained for the new correlation?

#### r = 0.40

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

Proportions in both the left-hand and right-hand tails tend to be relatively small.

Last week, Sarah had exams in her math and Spanish classes. On the math exam, the mean was µ = 30 with σ = 5, and Sarah had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 8, and Sarah had a score of X = 68. For which class should Sarah expect the better grade relative to her peers in each class?

Math

A population has µ = 50. Which value of σ would make X = 55 an extreme value out in the tail of the population distribution?

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