Exam 2 Statistics Psychology
What is the range for the set of scores: 7, 8, 11, 15, 17, 20?
13 or 14
What is the z-score for a sample mean of M = 21 where the population mean is 24, the population standard deviation is 3, and the sample size is 16?
-4
A beetle population has a mean length of 2.3 inches with standard deviation of 0.2 inches. What is the standard error for a sample size of 16?
0.05
A population of birth weights has a mean of μ = 8.1, and a standard deviation of σ = 0.1. What is the probability of an individual weight being greater than 8.4?
0.13%
There are 12 blue marbles, 8 red marbles, and 20 green marbles in a jar. A marbles is drawn from the jar and replaced. This is repeated 4 times, and each time a green marble is drawn. On the fifth time, what is the probability of drawing a green marble?
0.5
For a normal distribution with μ = 2 and σ = 0.5, what is p(X < 1.7) + p(X < 2.3)?
1
If the standard error among sample means is small, which of the following is true? I. All the possible sample means are clustered close together. II. A researcher can be confident that any individual sample mean will provide a reliable measure of the population. III. A researcher must be concerned that a different sample could produce a different conclusion.
1 and 2 only
Which of the following is a valid formula for calculating the sum of square deviations? I. SS= ∑(X - μ)^2 II. SS= ∑ X^2 - ((∑ X)^2/N) III. SS= ∑ X^2 - ((∑ X^2)/N)
1 and 2 only
Which of the following is equal to p(z > 2.00) in a normal distribution?I. p(z < -2.00) II. 1 - p(z < 2.00) III. ½ p(z > 1.00)
1 and 2 only
The wingspan of a population of insects is normally distributed with μ = 42 and σ = 4. Which of the following is true? I. It is possible for an insect in this population to have a wingspan of 30. II. It is likely for an insect in this population to have a wingspan of 30. III. It is unlikely an insect in this population to have a wingspan of 30.
1 and 3 only
Which of the following is a proper way to describe the probability of flipping heads on a fair coin? I. ½ II. 0.5 III. 50%
1, 2, and 3
Which of the following is a true statement for any population with mean μ and standard deviation σ? I. The distribution of sample means for sample size n will have a mean of μ. II. The distribution of sample means for sample size n will have a standard deviation of σ/square root of n. III. The distribution of sample means will approach a normal distribution as n approaches infinity.
1, 2, and 3
Which z-score mark separates the bottom 97.5% from the top 2.5% in a normal distribution?
1.96
If a multiple choice quiz has 100 questions in which every question has 4 choices, what is the probability of getting fewer than 20 questions correct merely by guessing?
10%
A researcher was asked to provide the standard deviation for a sample of fish weights. The researcher originally reported a standard deviation of 12.3 grams. However, upon closer investigation, she discovered that her scale was 1.1 grams off, and that each fish was actually 1.1 grams heavier than originally reported. What is the new standard deviation?
12.3
A researcher selects all of the possible samples with n = 8 scores from a population and computes the mean, dividing by n, for each sample. If the population mean is μ = 14, then what is the average value for all of the sample means?
14
A researcher is working with a population of data. He needs the standard distance between sample mean and the population mean to be at most 1.3. If the standard deviation of the population is 5, how large will the sample size need to be?
15
The population distribution of scores on a test is normal with μ = 40 and σ = 2. What is the probability of scoring less than a 38 on this test?
15.87%
The following is a sample of IQ's from a population. Find the sample standard deviation to two decimal places. 105, 107, 107, 120, 122, 147
15.95
Which of the following sets of scores has the least variability?
16, 17, 18, 19, 20
A population has a mean of 120 and a standard deviation of 10. If a sample of size 25 is collected, how much distance on average is expected between the sample mean and the population mean?
2
A scientist collects a sample of honeybees from a hive. He weighs the honeybees in grams and calculates the standard deviation to be 0.002 grams. If there are 1000 milligrams in one gram, what is the standard deviation in milligrams?
2
The population of scores on a nationally standardized test forms a normal distribution with μ = 300 and σ = 50. If you take a random sample of n = 25 students, what is the probability that the sample mean will be less than M = 280?
2.28%
The number of minutes spent reading in one day is normally distributed with a mean of μ = 23 and a standard deviation of σ = 5. How much time would you have to spend reading in a day to find yourself in the top 10% of readers?
29.4
A population of insect weights is normal with a mean of μ = 16. If the probability of a weight falling between 10 and 22 is 95.44%, what is the standard deviation for this distribution?
3
A sample has five scores, but only four of them are known. However, the sample standard deviation is s = 3, and the mean is M = 4. If the four known scores are 1, 1, 5, and 8, which is a possibility for the fifth score?
3 only
A teacher has given a test to a group of students. The set of scores is: 78, 79, 79, 81, and 83. Find the population variance for this group of tests.
3.2
For a normal population the sample mean M = 35 has the same z-score as the sample mean M = 43. What is the population mean?
39
Which of the following is not a valid probability?
6/5
What is the probability of a z-score falling within one standard deviation of the mean in a normal distribution?
68.26%
A scientist collected a sample of data of cactus heights. The data was rounded to the nearest half foot. If the minimum score was 3.5 feet and the range was 4 feet, what was the maximum score?
7 feet
If all the possible random samples of size n = 7 are selected from a population with μ = 70 and σ = 5 and the mean is computed for each sample, then what value will be obtained for the mean of all the sample means?
70
When Lorenzo finished his exam last week, he thought the test was over. But the instructor put z-scores on each student's paper and asked them to figure out their original score. The mean for the class is µ = 61 and standard deviation is σ = 8. Lorenzo's z-score is +1.75. What did he score on the exam?
75
A population of plant heights has a mean of μ = 13 and standard deviation of σ = 2. What is the probability of an individual plant having a height between 10 and 16?
86.64%
A researcher selects all of the possible samples with n = 7 scores from a population and computes the sample variance, dividing by n - 1, for each sample. If the population variance is σ2 = 9, then what is the average value for all of the sample variances?
9
The Central Limit Theorem
A mathematical theorem that specifies the characteristics of the distribution of sample means. For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of s/square root of n and will approach a normal distribution as n approaches infinity. Distribution of sample means for any population, no matter what shape, mean, or standard deviation. "Approaches" a normal distribution very rapidly; by the time the sample size reaches n = 30, the distribution is almost perfectly normal.
Biased
A sample statistic is this if the average value of the statistic either underestimates or overestimates the corresponding population parameter.
Unbiased
A sample statistic is this if the average value of the statistic is equal to the population parameter. The average value of the statistic is obtained from all the possible samples for a specific sample size, n. A statistic that, on average, provides an accurate estimate of the corresponding population parameter. The sample mean and sample variance are this.
Sampling Distribution
A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
Sampling with Replacement
A sampling technique that returns the current selection to the population before the next selection is made. A required part of random sampling.
Standardized Score
A score that has been transformed into a standard form.
Z-Score Transformation
A transformation that changes raw scores (X values) into z-scores.
Looking Ahead to Inferential Statistics
A typical research study begins with a question about how a treatment will affect the individuals in a population. Because it is usually impossible to study an entire population, the researcher selects a sample and administers the treatment to the individuals in the sample. To evaluate the effect of the treatment, the researcher simply compares the treated sample with the original population. If the individuals in the sample are noticeably different from the individuals in the original population, the researcher has evidence that the treatment has had an effect. On the other hand, if the sample is not noticeably different from the original population, it would appear that the treatment has no effect.
Z-Scores and Location
A z-score specifies the precise location of each X value within a distribution. The sign of the z-score (+ or −) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and μ.
Transformations of Scale
Adding a constant to each score does not change the standard deviation. Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant.
Other Standardized Distributions Based on z-Scores
Although z-score distributions have distinct advantages, many people find them cumbersome because they contain negative values and decimals. For this reason, it is common to standardize a distribution by transforming the scores into a new distribution with a predetermined mean and standard deviation that are whole round numbers. Such standardized scores are frequently used in psychological or educational testing. An instructor gives an exam to a psychology class. For this exam, the distribution of raw scores has a mean of μ = 57 with σ = 14. The instructor would like to simplify the distribution by transforming all scores into a new, standardized distribution with μ = 50 and σ = 10. To demonstrate this process, we will consider what happens to two specific students: Maria, who has a raw score of X = 64 in the original distribution; and Joe, whose original raw score is X = 43.
Computing z-Scores for Samples
Although z-scores are most commonly used in the context of a population, the same principles can be used to identify individual locations within a sample. The definition of a z-score is the same for a sample as for a population, provided that you use the sample mean and the sample standard deviation to specify each z-score location.
Independent Random Sample
An independent random sample requires that each individual has an equal chance of being selected and that the probability of being selected stays constant from one selection to the next if more than one individual is selected.
Probability and the Binomial Distribution
Binomial distributions are formed by a series of observations for which there are exactly two possible outcomes. For example, 100 coin tosses each yielding either a head or a tail. The two outcomes are identified as A and B, with probabilities of p(A) = p and p(B) = q (p + q = 1.00). The binomial distribution shows the probability for each value of X, where X is the number of occurrences of A in a series of n observations. When pn and qn are both greater than 10, the binomial distribution is closely approximated by a normal distribution with a mean of μ = pn and a standard deviation of σ = square root of npq. In this situation, a z-score can be computed for each value of X and the unit normal table can be used to determine probabilities for specific outcomes.
What is the best explanation for why a normal distribution is only an approximation for a binomial distribution?
Binomial values are discrete, and normal values are continuous.
When is the distribution of sample means identical to the population distribution?
Both a and b: When n = 1; and When the standard error equals the population standard deviation
Professor Chao's engineering final exam has a mean of µ = 79 and σ = 7. After standardizing the exam scores, the mean is µ = 80 and σ = 5. How did Professor Chao arrive at these new standardized scores for the exam?
Deciding what values will be simple to calculate
Sum of Squared Deviation Formulas
Definitional formula: SS = ∑(X - µ)^2 •Find each deviation score (X - µ). •Square each deviation score, (X - µ)^2. •Add the squared deviations Computational formula: SS = ∑X^2 - ((∑X)^2/N) Calculating SS for a sample is exactly the same as for a population, except for minor changes in notation. After you compute SS, however, it becomes critical to differentiate between samples and populations.
Standard Deviation and Descriptive Statistics
Describing an entire distribution: Rather than listing all of the individual scores in a distribution, research reports typically summarize the data by reporting only the mean and the standard deviation. Describing the location of individual scores: The mean and the standard deviation can be used to reconstruct the underlying scale of measurement (the X values along the horizontal line); The scale of measurement helps complete the picture of the entire distribution and helps to relate each individual score to the rest of the group.
As part of a clinical research project, a sample population has been using an experimental drug intended to improve memory. Study participants completed a comprehensive memory test following six weeks of treatment. The results were standardized and compared to mean results for the full population. Which of the following participants indicates promising results for this drug treatment?
Ellie, a 52-year-old woman with a memory test z-score of +2.75
In the context of inferential statistics, the variance that exists in a set of sample data is often classified as ______________.
Error Variance
Degrees of Freedom (df)
For a sample of n scores, use this for the sample variance are defined as df = n - 1. This is used to determine the number of scores in the sample that are independent and free to vary.
Probability
For a situation in which several different outcomes are possible, the probability for any specific outcome is defined as a fraction or a proportion of all the possible outcomes. If the possible outcomes are identified as A, B, C, D, and so on, then probability of A = number of outcomes classified as A/total number of possible outcomes.
Random Sampling
For the definition of probability used here to be accurate, it is necessary that the outcomes be obtained by a process called random sampling. A random sample requires that each individual in the population has an equal chance of being selected. A sample obtained by this process is called a simple random sample.
Standardizing a Sample Distribution
If all the scores in a sample are transformed into z-scores, the result is a sample of z-scores. The transformed distribution of z-scores will have the same properties that exist when a population of X values is transformed into z-scores. 1. The sample will have the same shape as the original sample. 2. The sample will have a mean of Mz = 0. 3. The sample will have a standard deviation of sz = 1.
Presenting the Mean and Standard Deviation in a Graph
In frequency distribution graphs, we identify the position of the mean by drawing a vertical line and labeling it with µ or M. Because the standard deviation measures distance from the mean, it is represented by a line or an arrow drawn from the mean outward for a distance equal to the standard deviation and labeled with a σ or an s.
Determining a Raw Score (X) from a Z-Score
In general, it is easier to use the definition of a z-score, rather than a formula, when you are changing z-scores into X values. X = μ + zσ
The 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 obtained from a population.
Probabilities & Proportions for Scores from a Normal Distribution
In many situations, it is necessary to find probabilities for specific X values rather than for z-scores. This requires two steps: Transform the X values into z-scores; and Use the unit normal table to look up the proportions corresponding to the z-score values. What is the probability of randomly selecting an individual with an IQ score less than 120? Restated in terms of proportions, we want to find the proportion of the IQ distribution that corresponds to scores less than 120.This requires two steps: Transform the X values into z-scores: z = (X - μ)/σ = (120 - 100)/15 = 20/15 = 1.33. Look up the z-score value in the unit normal table. Because we want the proportion of the distribution in the body to the left of X = 120, the answer will be found in column B.
Other Relationships between z, X, µ, and σ
In most cases, we simply transform scores (X values) into z-scores, or change z-scores back into X values. However, you should realize that a z-score establishes a relationship between the score, mean, and standard deviation. This relationship can be used to answer a variety of different questions about scores and the distributions in which they are located.
A distribution has µ = 50 and σ = 6. In a graphical representation of the distribution, where would the score X = 20 appear?
In the left tail of the curve
Variance and Inferential Statistics
In very general terms, the goal of inferential statistics is to detect meaningful and significant patterns in research results. Variability plays an important role in the inferential process because the variability in the data influences how easy it is to see patterns. In general, low variability means that existing patterns can be seen clearly, whereas high variability tends to obscure any patterns that might exist.
If a researcher is concerned that a standard error is too big for the sample mean to provide a reliable measure of the population mean, what can the researcher do?
Increase the sample size.
Variance
It equals the mean of the squared deviations. It is the average squared distance from the mean.
Standardized Distribution
It is composed of scores that have been transformed to create predetermined values for μ and σ. It is used to make dissimilar distributions comparable.
Deviation
It is distance from the mean.
What happens to the standard distance between the sample mean and the population mean when the sample size is multiplied by 4?
It is divided by 2.
Using z-Scores to Standardize a Distribution
It is possible to transform every X value in a distribution into a corresponding z-score. If every X value is transformed into a z-score, then the distribution of z-scores will have the following properties: The distribution of z-scores will have exactly the same shape as the original distribution of scores; The z-score distribution will always have a mean of zero; The distribution of z-scores will always have a standard deviation of 1. A standardized distribution is composed of scores that have been transformed to create predetermined values for μ and σ. Standardized distributions are used to make dissimilar distributions comparable.
Sample Standard Deviation (s)
It is represented by the symbol s and equal the square root of the sample variance.
Sample Variance (s^2)
It is represented by the symbol s^2 and equals the mean squared distance from the mean. It is obtained by dividing the sum of squares by (n - 1). The sum of the squared deviations divided by df = n-1. An unbiased estimate of the population variance.
Population Standard Deviation (σ)
It is represented by the symbol σ and equals the square root of the population variance.
Population Variance (σ^2 )
It is represented by the symbol σ^2 and equals the mean squared distance from the mean. It is obtained by dividing the sum of squares by N. The average squared distance from the mean; the mean of the squared deviations.
The Range
It is the distance covered by the scores in a distribution, from the smallest score to the largest score.One commonly used definition of this simply measures the difference between the largest score (Xmax) and the smallest score (Xmin). When the scores are measurements of a continuous variable, this can be defined as the difference between the upper real limit (URL) for the largest score (Xmax) and the lower real limit (LRL) for the smallest score (Xmin).
Standard Deviation
It is the square root of the variance and provides a measure of the standard, or average distance from the mean.
SS or Sum of Squares
It is the sum of the squared deviation scores.
Variability
It provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered together. It describes the distribution. It measures how well an individual score (or group of scores) represents the entire distribution.
All of the possible random samples of size 5 are selected from a population and the variance among these sample means is 16. If all possible random samples of size 10 are selected from the same population, what can we say about the variance of this new set of sample means?
It will be less than 16.
There are five cards face down on a table with the values 1, 2, 3, 4, and 5 written on them. James says that the probability of randomly turning over an even number is ½ because there are two possibilities: even and odd. Which of the following statements is true?
James is incorrect because the two possibilities of "even" and "odd" are not equally likely in this situation.
What is the probability of a z-score being less than 2 standard deviations from the mean in a normal distribution?
More than 95%
The standard error for a particular sample is 3.6. A researcher needs the standard error to be 1.2. What should the researcher do to the sample size?
Multiply the sample size by 9.
Using z-Scores for Making Comparisons
One advantage of standardizing distributions is that it makes it possible to compare different scores or different individuals even though they come from completely different distributions. Normally, if two scores come from different distributions, it is impossible to make any direct comparison between them. Using z-scores makes such comparisons possible.
Looking Ahead to Inferential Statistics 2
Probability forms a direct link between samples and the populations from which they come. The research begins with a population that forms a normal distribution with a mean of μ = 400 and a standard deviation of σ = 20. A sample is selected from the population and a treatment is administered to the sample. The goal for the study is to evaluate the effect of the treatment. To determine whether the treatment has an effect, the researcher simply compares the treated sample with the original population. If the individuals in the sample have scores around 400 (the original population mean), then we must conclude that the treatment appears to have no effect. On the other hand, if the treated individuals have scores that are noticeably different from 400, then the researcher has evidence that the treatment does have an effect. The problem for the researcher is determining exactly what is meant by "noticeably different" from 400. Probability helps researchers define "noticeably different."
The Role of Probability in Inferential Statistics
Probability is used to predict the type of samples that are likely to be obtained from a population. Thus, probability establishes a connection between samples and populations. Inferential statistics rely on this connection when they use sample data as the basis for making conclusions about populations.
Measuring Variance and Standard Deviation for a Population
Recall that variance is defined as the mean of the squared deviations. First find the sum, and then divide by the number of scores. It is identified by the notation SS (for sum of squared deviations), and is referred to as the sum of squares. There are two algebraically equivalent formulas for SS. They yield the same value but they look different and are used in different situations.
Which of the following is a true statement?
Samples tend to be less variable than their populations.
Sampling Error
Sampling error is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
The distribution for scores on a history exam has µ = 70 and σ = 4. The distribution for a set of scores on a sociology exam has µ = 68 and σ = 4. Tenisha scored 76 on both exams. Which of the following statements best describes the grades Tenisha will receive on her exams?
She will receive a higher grade on the sociology exam based on standardization.
The Mean of the Distribution of Sample Means
The average value of all the sample means is exactly equal to the value of the population mean. The formal statement of this phenomenon is that the mean of the distribution of sample means always is identical to the population mean. This mean value is called the expected value of M. The sample mean is an example of an unbiased statistic, which means that on average the sample statistic produces a value that is exactly equal to the corresponding population parameter. In this case, the average value of all the sample means is exactly equal to μ.
Deviation Score
The distance (and direction) from the mean to a specific score.
The Shape of the Distribution of Sample Mean
The distribution is almost perfectly normal if either of the following two conditions is satisfied: The population from which the samples are selected is a normal distribution; and The number of scores (n) in each sample is relatively large, around 30 or more.
More About Standard Error
The general concept of sampling error is that a sample typically will not provide a perfectly accurate representation of its population. More specifically, there typically is some discrepancy (or error) between a statistic computed for a sample and the corresponding parameter for the population. For each individual sample, you can measure the error (or distance) between the sample mean and the population mean. For some samples, the error will be relatively small, but for other samples, the error will be relatively large. The standard error provides a way to measure the "average", or standard, distance between a sample mean and the population mean.
Measuring Standard Deviation and Variance for a Sample
The goal of inferential statistics is to use the limited information from samples to draw general conclusions about populations. The basic assumption of this process is that samples should be representative of the populations from which they come. This assumption poses a special problem for variability because samples consistently tend to be less variable than their populations. Fortunately, the bias in sample variability is consistent and predictable, which means it can be corrected.
Law of Large Numbers
The law of large numbers states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean.
What is necessary to determine the z-score for a raw score in a particular set of scores?
The mean and standard deviation for the set
Expected Value of M
The mean of the distribution of sample means is equal to the mean of the population of scores, μ, and is called the expected value of M.
What is a sampling error?
The natural error that exists between a sample and its corresponding population
What is the benefit of converting raw scores from a sample into z-scores?
The new scores are easier to compare.
Probability and the Normal Distribution
The normal distribution is symmetrical, with the highest frequency in the middle and frequencies tapering off as you move toward either extreme. Because the locations in the distribution are identified by z-scores, the percentages shown in the figure apply to any normal distribution regardless of the values for the mean and the standard deviation.
Percentiles and Percentile Ranks
The percentile rank for a specific X value is the percentage of individuals with scores at or below that value. When a score is referred to by its rank, the score is called a percentile. The percentile rank for a score in a normal distribution is simply the proportion to the left of the score.
Probability and the Distribution of Sample Means
The primary use of the distribution of sample means is to find the probability associated with any specific sample. Recall that probability is equivalent to proportion. Because the distribution of sample means presents the entire set of all possible sample means, we can use proportions of this distribution to determine probabilities. The population of scores on the SAT forms a normal distribution with μ = 500 and σ = 100. If you take a random sample of n = 16 students, what is the probability that the sample mean will be greater than M = 525? Restate this probability question as a proportion question: Out of all the possible sample means, what proportion has values greater than 525? "All the possible sample means" refers to the distribution of sample means. The problem is to find a specific portion of this distribution. Because of the central limit theorem, we know: 1. The distribution is normal because the population of SAT scores is normal. 2. The distribution has a mean of 500 because the population mean is μ = 500. 3. For n = 16, the distribution has a standard error of σM = 25: = s/square root of n = 100 / square root of 16 = 100/4 = 25 The value 525 is located above the mean by 25 points, which is exactly 1 standard deviation (in this case, exactly 1 standard error). Thus, the z-score for M = 525 is z = +1.00. The unit normal table indicates that 0.1587 of the distribution is located in the tail of the distribution beyond z = +1.00. Our conclusion is that it is relatively unlikely, p = 0.1587 (15.87%), to obtain a random sample of n = 16 students with an average SAT score greater than 525.
Purposes of z-Scores
The purpose of z-scores, or standard scores, is to identify and describe the exact location of each score in a distribution. Suppose you received a score of X = 76 on a statistics exam. How did you do? Your score of X = 76 could be one of the best scores, or it might be the lowest score. To find the location of your score, you must have information about the other scores in the distribution. A second purpose for z-scores is to standardize an entire distribution. A common example of a standardized distribution is the distribution of IQ scores. Although there are several different tests for measuring IQ, the tests usually are standardized so that they have a mean of 100 and a standard deviation of 15.
Which of the following is not a characteristic of the distribution of sample means?
The sample means should have similar standard deviations as the population standard deviation.
Characteristics of the Distribution of Sample Mean
The sample means should pile up around the population mean. The pile of sample means should tend to form a normal-shaped distribution. In general, the larger the sample size, the closer the sample means should be to the population mean, μ.
Probability and Frequency Distributions
The situations in which we are concerned with probability usually involve a population of scores that can be displayed in a frequency distribution graph. If you think of the graph as representing the entire population, then different portions of the graph represent different portions of the population. Because probabilities and proportions are equivalent, a particular portion of the graph corresponds to a particular probability.
Standard Deviation Formula
The square root of the variance
The Standard Error of M
The standard deviation of the distribution of sample means, σM, is called the standard error of M. The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ). The standard error describes the distribution of sample means. Measures how well an individual sample mean represents the entire distribution, specifically, how much distance is reasonable to expect between a sample mean and the overall mean for the distribution of sample means The magnitude of the standard error is determined by two factors: The size of the sample: The law of large numbers states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean; and The standard deviation of the population from which the sample is selected: there is an inverse relationship between the sample size and the standard error. Bigger samples have smaller error, and smaller samples have bigger error.
When computing z for a sample mean, which quantity is used?
The standard error
Which quantity decreases as the sample size increases?
The standard error
The Unit Normal Table
The unit normal table lists several different proportions corresponding to each z-score location. Column A of the table lists z-score values. For each z-score location, columns B and C list the proportions in the body and tail, respectively. Finally, column D lists the proportion between the mean and the z-score location. Because probability is equivalent to proportion, the table values can also be used to determine probabilities. A table listing proportions corresponding to each z-score location in a normal distribution.
Z-Score Formula
The z-score definition is adequate for transforming back and forth from X values to z-scores as long as the arithmetic is easy to do in your head. For more complicated values, it is best to have an equation to help structure the calculations. . z = (X - μ)/σ
Which of the following is true for the quantity ∑(X - μ)?
This is the sum of the deviations, and it is always 0.
Z-Score
This specifies the precise location of each X value with a distribution. The sign of the z-score (+ or -) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations.
Marisol received a score of X = 70 on her chemistry exam and a score of X = 59 on her geography exam. The mean for the chemistry exam is µ = 66, and the mean for the geography exam is µ = 50. On which exam will Marisol receive a higher grade?
Unknown; the standard deviation for each set of exam scores is required.
Binomial Distribution
Using the notation presented here, the binomial distribution shows the probability associated with each value of X from X=0 to X=n.
Variance Formula (Population)
Variance = mean squared deviation = sum of squared deviations/number of scores = SS/N
Probabilities, Proportions, and z-Scores
What proportion of the normal distribution corresponds to z-score values greater than z = 1.00? First, you should sketch the distribution and shade in the area you are trying to determine. In this case, the shaded portion is the tail of the distribution beyond z = 1.00. To find this shaded area, you simply look for z = 1.00 in column A to find the appropriate row in the unit normal table. Then scan across the row to column C (tail) to find the proportion. For a normal distribution, what z-score separates the top 10% from the remainder of the distribution? To answer this question, sketch a normal distribution and draw a vertical line that separates the highest 10% (approximately) from the rest. To find the z-score value, you simply locate the row in the unit normal table that has 0.1000 in column C or 0.9000 in column B.
When is the standard deviation of a set of scores equals to 0?
When the range of the scores is 0
Under what circumstances would a score that is below the mean by 3 points appear to be very far from the mean?
When the standard deviation is much less than 3.
When is the population standard deviation greater than the population variance?
When the variance is less than 1
Samples, Populations, and the Distribution of Sample Means
Whenever a score is selected from a population, you should be able to compute a z-score that describes exactly where the score is located in the distribution. If the population is normal, you also should be able to determine the probability value for obtaining any individual score. However, the z-scores and probabilities that we have considered so far are limited to situations in which the sample consists of a single score. In general, the difficulty of working with samples is that a sample provides an incomplete picture of the population.
Sample Variability and Degrees of Freedom
With a population, you find the deviation for each score by measuring its distance from the population mean. With a sample, the value of m is unknown and you must measure distances from the sample mean. Because the value of the sample mean varies from one sample to another, you must first compute the sample mean. However, calculating the value of M places a restriction on the variability of the scores in the sample. In general, when a sample has n scores, the first (n - 1) scores are free to vary, but the final score is restricted.
Z-Scores and Location within the Distribution of Sample Means
Within the distribution of sample means, the location of each sample mean can be specified by a z-score: z = (M - μ)/ σM The sign tells whether the location is above (+) or below (-) the mean. The number tells the distance between the location and the mean in terms of the number of standard deviations.
For a normal distribution with a mean of 60 and a standard deviation of 5, what X values separate the middle 95% from the extreme values?
X = 50.2 and X = 69.8
A distribution with µ = 61 and σ = 6 has been standardized to reflect a new mean of µ = 50 and a standard deviation of σ = 10. What is the new standardized score for a score of X = 52 in the original distribution?
X= 35
For a population with µ = 69 and σ = 4, which of the following X scores will convert to a positive z-score of magnitude greater than 1.0?
X= 74
Deviation Score Formula
deviation score = X - µ
A random sample is obtained from a population with μ = 120 and σ = 20, and a treatment is administered to the sample. Which of the following outcomes would be considered noticeably different from a typical sample that did not receive the treatment?
n = 144 with M = 124
Given that p = 0.5, for which value of n will the bar graph for the corresponding binomial distribution look most like a normal distribution?
n= 100
For samples selected from a population with μ = 90 and σ = 30, what sample size is necessary to make the standard distance between the sample mean and the population mean equal to 5 points?
n= 36
For a binomial distribution with p = 0.75 and n = 15, the probability of A occurring more than 10 times is about 68.65%. Using a normal distribution with μ = 11.25, σ = 1.667, and p(X >10.5) to approximate this probability gives 67.26%. Which of the following best explains the discrepancy between the actual value of 68.65% and the approximation of 67.26%?
nq is not high enough to achieve a closer approximation.
Range Formulas
range = Xmax - Xmin range = URL for Xmax - LRL for Xmin
An unbiased statistic is one in which ...
the average value of the statistic is equal to the population parameter.
A distribution of exam scores has a mean of µ = 52 and standard deviation of σ = 6. What is the z-score for the exam score X = 61?
z = +1.5
A calculus exam has a mean of µ = 73 and a standard deviation of σ = 4. Trina's score on the exam was 79, giving her a z-score of +1.50. The teacher standardized the exam distribution to a new mean of µ = 70 and standard deviation of σ = 5. What is Trina's z-score for the standardized distribution of the calculus exam?
z= +1.50
A sample has a mean of M = 64 and standard deviation of s = 3. What is the z-score for a sample score of X = 70?
z= +2.00
Which of the following z-scores indicates an X value that falls farthest below the mean for the distribution?
z= -1.50
A researcher has compiled a set of scores with a mean of µ = 39 and standard deviation of σ = 4. What is the z-score for the raw score X = 28?
z= -2.75
A distribution of scores for a test of life stressors has a mean of µ = 125 and standard deviation of σ = 15. The researcher calculates z-scores to standardize the distribution. What are the mean and the standard deviation for the z-scores in this distribution?
µ = 0, σ = 1
Within a population having standard deviation of σ = 20, the raw score X = 75 has a z-score of -1.25. What is the mean for this population?
µ = 100
In a distribution, a score of X = 60 has a z-score of -3.0. A score of X = 85 has a z-score of +2.0. What are the mean and the standard deviation for this population?
µ = 75, σ = 5
Each score in a set of data is multiplied by 5, and then 7 is added to the result. If the original mean is 8 and the original standard deviation is 2, what are the new mean and new standard deviation?
μ = 47, σ = 10
A distribution of 1000 test scores has a mean of µ = 287. Following standardization, what is the standard deviation for this distribution?
σ = 1
A test of life satisfaction has a mean of µ = 60. Jenna's test score of X = 72 has a z-score of +3.0. What is the standard deviation for this distribution of test scores?
σ = 4
Find the population variance and standard deviation for the following ages of children in a family: 1, 3, 5, 7, 8, 11, 14.
σ = 4.17, σ2 = 17.43