Statistics Exam 2
There is no ___ in the selection process and you cannot apply the definition of ___ to situations where the possible outcomes are not likely
bias; probability
Using the notation presented here, the ___ ___ shows the probability associated with each value of x from x = 0 to x = n
binomial distribution
The ___ ___ tends to approximate a normal distribution, particularly when n is ___
binomial distribution; large
The ____ distribution will be a nearly perfect normal distribution when __ and __ are both equal to or greater than __
binomial; pn and qn; 10
A z-score value near zero indicates a ___ ___ ___; a z value beyond + 2.00 or - 2.00 indicates an ___ ___
central representative sample;extreme sample
The extremely unlikely values provide us with ___ for determining whether or not a treatment has caused individuals to be ____
criteria; different
Probability forms a ___ ___ between samples and the populations from which they come
direct link
The exact shape of the normal distribution is specified by an ____ relating each X value (score) with each Y value (frequency)
equation
A z-score of ±2.0 is an ___ score in the tail of the distribution and has a ___ of only p=.0228
extreme;probability
The role of ___ ___ is to use the sample data as the basis for answering questions about the population
inferential statistics
To gain maximum accuracy when using the normal approximation, you must remember that each X value in the binomial distribution is actually an ___, not a point on the scale
interval
The body always corresponds to the ____ part of the distribution whether it is on the right or left hand side
larger
The frequency tapers off as you move farther from the ____ in either direction
middle
A sample that falls beyond z = ±1.96 is
not only extreme, it is also extremely unlikely. Thus, it appears the treatment had an effect
Researchers want to know if there is a ____ ____ between the population and the sample to determine if a treatment had an effect
noticeable difference
Probability of Selecting A king
p (king) = 4/52 = 1/13
Probability of Selecting A spade
p (spade) = 13/52 = 1/4
In a set of 100 samples, the probability of obtaining any specific samples is 1 out of 100 or
p = _1_ 100
Probability of selecting a king of hearts
p= 1/52
Binomial distribution for the number of heads obtained in 2 coin tosses
p=p(heads) = 1/2 q=p(heads) = 1/2
Large samples will be representative of the ____ from which they are selected
population
By knowing the makeup of a _____, we can determine the _____ of obtaining specific samples
population; probability
To keep the ____ from changing from one selection to the next, it is necessary to replace each ____ before you make the next selection
probabilities; sample
For a situation in which several different outcomes are possible, the _____ of any specific outcome is defined as a fraction or a proportion of all the possible outcomes
probability
It is possible to define a normal distribution in terms of its ____; that is, a distribution is normal if and only if it has all the right ____. (z= 1 makes up 34.13% of the scores, z=2 makes up 13.59%)
proportions; proportions
Most people tend to focus on the sample ____ and pay very little attention to sample ___
proportions;size
A ___ ___ requires that each individual in the population has a equal chance of being selected
random sample
Sample size is one of the primary considerations in determining how well a sample mean ____ the sample as a whole
represents
Inferential statistics rely on this connection when they use ____ ____ as the basis for making conclusions about ____
sample data; populations
It is the collection of ___ ___ for all the possible random samples of a particular size (n) that can be obtained from a population
sample means
A distribution of statistics obtained by selecting all the possible samples of a specific size from a population
sampling distribution
When statistics are obtained from samples, it is referred to as a ___ ___
sampling distribution
A sample provides an incomplete picture of the population. This difference, or error between sample statistics and the corresponding population parameters, is called ___ ___
sampling error
The discrepancy or amount of error between a sample statistic and its corresponding population parameter
sampling error
Whenever a ___ is selected from a population, you should be able to compute a z-score that describes exactly where the score is ___ in the distribution
score;located
it will be possible to describe how any ___ sample is related to all the other possible ___
specific;samples
A second requirement, necessary for many statistical formulas, states that the probabilities must ___ ___ from one selection to the next if more than one individual is selected
stay constant
Because the normal distribution is ____, the proportions of the right hand side are exactly the same as the corresponding proportions on the left-hand side
symmetrical
sections on the left side of the distribution have exactly the same areas as the corresponding sections on the right side because the normal distribution is ____
symmetrical
The normal distribution is ____with a single ___ in the middle
symmetrical ; mode
With a small sample, you risk obtaining a ____ sample
unrepresentative
If the population is normal, you should be able to determine the probability ___ for obtaining any ___ ___
value;individual score
Although the ____ will change signs (+ and -) from one side to the other, the proportions will always be ____
z- scores; positive
Boundaries are set through ___
z-scores
The binomial distribution tends to be normal in shape; thus, we can compute probability values directly from ___ and the ___ ___ ___
z-scores ; unit normal table
Because the locations in the distribution are identified by _____, the percentages shown apply to any normal distribution regardless of the values for the mean and ___ ___
z-scores; standard distribution
A large sample should be more accurate than a small sample. In general, as the sample size increases, the error between the sample mean and the population mean should decrease
...
A z-score can describe the position of any specific sample within the distribution of sample means
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As the sample size increases beyond n = 1, the sample becomes a more accurate representative of the population and the standard error decreases
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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 σ/√n , and will approach a normal distribution as n approaches infinity
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In general, the larger the sample size, the closer the sample means should be to the population mean
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Knowing the standard error gives researchers a good indication of how accurately their sample data represents the population they are studying
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Means obtained with a large sample size should cluster relatively close to the population mean; the means obtained from small samples should be more widely scattered
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Standard error=σM = _σ_ √n
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The distribution of driving speeds is shown in figure 6.10 with the appropriate area shaded. Determine the z-scores for the X values: X=55: z = X-µ/σ = 55-58/10 = -3/10 =-.030 X=65: z = X-µ/σ = 65-58/10 = 7/10 = .070
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The distribution of sample means should be tightly clustered around µ for large samples and more widely scattered for smaller samples
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The highway department conducted a study measuring driving speeds on a section of highway. The average speed was µ=58 miles per hour with a standard deviation of σ=10. What proportion of cars are traveling between 55 and 65 miles per hour? p (55 < X < 65) = ?
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The larger the sample size (n), the more probable it is that the sample mean will be close to the population mean
...
The mean of the distribution of sample means is equal to µ
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The pile of sample means should tend to form a normal-shaped distribution
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The s.e. can be viewed as a measure of the reliability of a sample mean
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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
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The sample mean is not expected to give a perfectly accurate representation of the population mean; there will be some error, and the standard error tells exactly how much error, on average, should exist between the sample mean and the unknown population mean
...
The sample mean provides information about the value of the unknown population mean
...
The sample means should pile up around the population mean. They should be relatively close to the population mean
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The smallest sample and the biggest error occur when n=1. The means between the population and the sample are identical. When n=1, Standard error=σM =σ=standard deviation
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The standard deviation of the distribution of sample means is called the standard error of M
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The standard error measures the standard amount of difference between M and µ that is reasonable to expect simply by chance
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The standard error provides a way to measure the "average" or standard distance between a sample mean and the population mean
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The term reliability refers to the consistency of different measurement of the same thing and obtain identical (or nearly identical) values
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There is an inverse relationship between the sample size and the standard error: bigger samples have smaller error, and smaller samples have bigger error
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Using column D of the unit normal table, these two proportions are 0.1179 and 0.2580. The total proportion is obtained by adding these two sections. p(55 < X < 65)= p(-30 < z < +0.70)= 0.1179 + 0.2580) = .3759 Or, 37.6% of the cars travel between 55 and 65 miles per hour.
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Using the z-scores and the unit normal table, it is possible to find the probability associated with any specific sample mean
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What is the minimum score necessary to be in the top 15% of the SAT distribution
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What is the probability of obtaining 15 heads in 20 tosses of a balanced coin?
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What is the probability of randomly selecting an individual with an IQ of less than 130?
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When sample size is increased, the standard error gets smaller, and the sample means tend to approximate µ more closely
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standard error defines the relationship between sample size and the accuracy with which M represents µ
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standard error plays a very important role in inferential statistics. Because of its importance, it is often reported in scientific papers
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A z-score of ± 2.00 indicates that the sample mean is much larger than usually would be expected: It is greater than the expected value of M by twice the standard distance
1
Usually we set a ___ criteria determined by _____
95%;z = ±1.96
We refer to data as binomial when a variable is measured on a scale consisting of exactly two categories
Heads/tails Male/female
The percentage of the individuals in the distribution who have scores that are less than or equal to the specific score
Percentage Rank
If the possibility outcomes are identified as A, B, C, D and some, then
Probability of A = number of outcomes, classified as A/total number of possible outcomes
_____ is used to predict what kind of samples are likely to be obtained from a _____
Probability; population
The magnitude of the standard error is determined by
The size of the sample, The standard deviation of the population from which the sample is located.
Using driving speeds from the previous example, what proportion of cars travel between 65 and 75 miles per hour?
X=65: z = X-µ/σ = 65-58/10 = 7/10 = .070 X=75: z = X-µ/σ = 75-58/10 = 17/10 = 1.70
We use a modified formula to determine the z-score for a sample by using M in place of X and by using the standard error instead of the standard deviation
Z = M - µ/σM
