Weeks 3 and 4 Flashcards
A standard error is a standard deviation of a hypothesized population.
a) true
b) false
b) false
Explanation … A standard error is, by definition, a standard deviation of a sampling distribution, not of a hypothesized population.
A hypothesis test usually ends with a “p-value”. This value expresses the proportion of area in the tails of what distribution?
a) the population distribution
b) the sampling distribution
c) the sample distribution
d) the standard error
b) the sampling distribution
Explanation … In the logic of hypothesis testing you form a sampling distribution for some statistic under the hypothesis you have chosen. You then observe the value of that statistic in your experimental sample, and this picks out a point within the sampling distribution. The p-value is the proportion of the total area under the sampling distribution curve that lies farther out in the tails than this point. Thus the p-value is a feature of the sampling distribution.
For a given hypothesis and sampling distribution – the farther an observed sample statistic falls out in the tails of its sampling distribution, the larger the corresponding p-value.
a) true
b) false
b) false
Explanation … Sampling distributions show the probability of results to expected under a hypothesis. If the value of a statistic in an observed sample falls far out in the tails of its sampling distribution then there is only a small proportion of values that could be more extreme. Since the p-value is the same as this tiny proportion left out in the tails, you can see that the smaller the proportion, the smaller the p-value
The rate of Type I errors (called “alpha“ and usually set at 5% in psychology) that you are willing to hazard in a hypothesis test is be set before you actually take any data.
a) true
b) false
Explanation … A p-value reflects the proportion of the area under a sampling distribution that is more extreme than a value determined from experiment. A hypothesis test doesn’t end, however, after determining the p-value – you still have to see if the p-value is bigger or smaller than alpha. If p is smaller than alpha then you should reject the hypothesis and if it is larger you should fail to reject. While both p and alpha are similar in that they can be pictured as areas in the tails of a sampling distribution, there is a big difference operationally. The p-value is something that is locked to experimental results whereas the alpha-level is chosen ahead of time. To understand the alpha-level, you can think of the population hypothesis, its consequent sampling distribution, and the rejection region in the tails that is determined by a theoretical machinery of hypothesis testing that exists independent of any measurements. The p-value, in contrast, depends on the value of the statistic in the sample and so is changeable.
5) The plots in a) and b) below show means and standard deviations whereas in c) and d) they show means and standard errors. In which plot is it possible to say that the difference in means is definitely NOT statistically significant? (Hint, the standard error of the mean for a dataset is always smaller than the standard deviation)
a) first bar and error bar is much higher than second bar
b) first bar and error bar is higher than the second, but the second one has a much higher error bar that is closer to the first bar (both with larger error bars)
c)
like a
d) like a except both have much larger error bars)
d)
Explanation … We are looking for 2 means that are NOT significantly different from each other. In a z-test, however, 2 means ARE considered significantly different from each other if they are separated by more than 1.96 SE (assuming alpha = .05 and a 2-tailed test). This is certainly the case in diagram c), therefore this is not the right answer. Diagram d) does seem to depict the situation we are after since the means are clearly separated by only about 1.3 SE or so. However, let’s keep looking at other diagrams just to be sure. Diagram a) shows means and standard deviations. One can see that the means are more than 2 SD apart, but since the standard error of the mean for a dataset is always smaller than the standard deviation, these means must be at least 2 SE apart too (you should think about why this is true!). So diagram a) cannot be the right answer. Finally, in diagram b) for a large enough sample size the standard error will become sufficiently smaller than the standard deviation to make the difference in means significant. So b) isn’t the right answer either. We have therefore excluded all answers except answer d). This question relies on you realizing that the standard error of the mean for a dataset is always smaller than the standard deviation.
6) If you want to reject a hypothesis about a population, which would you rather have … a narrow sampling distribution or a wide sampling distribution?
a) narrow sampling distribution
b) wide sampling distribution
a) narrow sampling distribution
Explanation … You reject a hypothesis if the observed value of a statistic in a real sample ends up sufficiently far out in the tail of its sampling distribution. Conversely, you fail to reject a hypothesis if the sample statistic ends up somewhere in the middle of the sampling distribution rather than somewhere in the tail.
Now imagine what happens when you have the widest possible sampling distribution … something with tails that are practically at plus and minus infinity. In this case a sample with its calculated statistic will have a very hard time landing out in the tails and so it will be difficult to reject the hypothesis. But now imagine the narrowest possible sampling distribution where 95% of its area is concentrated within a tiny range of values. You can see that a sample statistic is much more likely to end up somewhere out in the tails of this sort of distribution since it is practically all tails in the first place. You are therefore much more likely to reject a hypothesis if you have this sort of sampling distribution.
Narrow sampling distributions thus increase the likelihood of rejecting a hypothesis whereas wide sampling distributions make it less likely. In statistics, we often purposely design experiments so that the sampling distribution will be as narrow as possible.
If you want to reject a hypothesis about a population, which would you rather have … a small standard error or a large one?
a) small standard error
b) large standard error
a) small standard error
Explanation … This is really the same question as the previous one. The standard error is a measure of the width of a sampling distribution. The smaller the standard error, the narrower the sampling distributions. And, as we saw before, the narrower the sampling distribution the easier it is to reject a hypothesis.
8) Which of the selections below best describes what goes in the bottom of the ratio that is constructed when testing a hypothesis?
a) the standard deviation of a sampling distribution
b) the standard deviation of the hypothesized population
c) the mean of a sampling distribution
d) the standard deviation of a sample
a) the standard deviation of a sampling distribution
Explanation … When testing a hypothesis, you first form a ratio, then compare the value of the ratio to a critical point in a known distribution. The numerator (top) of this ratio consists of the deviation of your observed sample statistic from the value that you expect for this statistic if the hypothesis regarding the nature of the population is true. For instance, if you hypothesize that the population has skew = 0 then you would expect that most samples should have skew near zero, so the top of the ratio would consist of the deviation between whatever the sample skew actually is and the expected value of 0. The bottom (denominator) of the ratio consists of the value of the standard deviation for the sampling distribution you are using. But the standard deviation of a sampling distribution has another name…the standard error. The ratio thus expresses how many standard errors what you actually measure in your sample deviates from what you expect to see.
The sampling distribution of the mean for the population distribution shown below is Normal (assume that n = 100).
a) true
b) false
a) true
Explanation … This is an application of the Central Limit Theorem (CLT). The CLT says that no matter what distribution a parent population might have, the sampling distribution of the mean for that population will be approximately normal if the sample size is large enough. Conventionally, statisticians say that a sample size of 30 or more is large enough for the CLT to guarantee that the sampling distribution of the mean will be reasonably close to an ideal normal distribution.
According to the Central Limit Theorem, the sampling distribution of the variance for the population distribution shown below is Normal (assume that n = 100).
c) true
d) false
b) false
Explanation … The Central Limit Theorem tells you about the sampling distribution of the mean. It doesn’t tell you anything about the sampling distribution of the variance, or the sampling distribution of the skew, or the range, or any other statistic. It may be that other sampling distributions turn out to be Normal when you have large sample sizes, but you can’t really tell unless you have specialized information about them (it so happens that the sampling distribution of the variance is not Normal, even for large sample sizes). Remember … the CLT is a theorem about the sampling distribution of the mean.
Following a 2-tail z-test with result z = 1, you should reject the hypothesis about the population (assume alpha = .05).
a) true
b) false
b) false
Explanation … After calculating a z-value you need to compare it to a standard normal distribution to see if the z-value is in the tails. Since the 2-tailed p = .05 level in a standard normal distribution is at ±1.96, the hypothesis of a z-test will only be rejected if z > 1.96 or z < -1.96. In the question, z = 1, so we should not reject the hypothesis, instead we should accept it.
12) Which of the following situations yields a larger standard error of the mean?
a) a population standard deviation of OX = 10 and a sample size of n = 25
b) a population standard deviation of OX = 15 and a sample size of n = 100
a) a population standard deviation of Ox = 10 and a sample size of n = 25
Explanation … The formula for the standard error of the mean is
OM = Ox/square root of n. This formula (together with a close relative) is the most often-used formula you will encounter in PSYC3000. Applying this in the 2 cases gives
a) OM = 10/square root of 25 = 10/5 = 2, and
b) OM = 15/square root of 100 = 15/10 = 1.5 . Thus, the situation in a) has the larger standard error.
Suppose that you administer the Brief Symptom Inventory (a self-report psychological questionnaire) to 25 experimental subjects. The results are expressed as standardized T-scores (normally distributed population, population mean = 50, population standard deviation = 10). You find that the mean Anxiety T-score in the sample is 47. What test would you use to find out if the participants in your sample are significantly less anxious than in the population of psychologically well-adjusted individuals)?
a) z-test
b) t-test
c) test of skew
d) test of kurtosis
a) z-test
Explanation … The “T-score” system used for anxiety scores in the Brief Symptom Inventory is a system that standardizes all its scores against the scores of a reference population of psychologically well-adjusted individuals. This means that the test has been administered to many such individuals in the past and the scoring has been adjusted so that in the reference population the mean is u = 50 and the standard deviation is OX = 10. In a test we would hypothesize that the sample comes from this known population. Since both the population standard deviation and mean are known, this will be a z-test.
What is the value of the standard error of the mean, M, in the question above about Anxiety T scores?
a) 0.25
b) -0.25
c) 1.96
d) 2
e) 3
d) 2
Explanation … The standard error of the mean depends on the properties of the population and the sample size. We know what the properties of the population are supposed to be because the developers of the Brief Symptom Inventory used a standardized T-scoring system to quantify their results. Thus, we know that a reference population of psychologically normal individuals should have Anxiety T-scores with u = 50 and OM =10. We can therefore calculate OM for this population and for a sample size of n = 25 …
OM = Ox/square root of n = 10/square root of 25 = 10/5 = 2.
In the question about Anxiety T scores above, what would have been the value of M if the mean score in the sample was 44 instead of 47?
a) 0.25
b) 0.50
c) 1
d) 2
e) 6
d) 2
Explanation … The value of the standard error is a consequences of choices we make before we ever measure anything. Its value is predetermined once we hypothesize a population and choose an experimental design (e.g., sample size). It usually doesn’t depend on actual measurements in a sample and, in particular, it doesn’t depend on the value of M (as you can see from its formula). So the answer to this question is the same as for the previous one.
An exception to this happens when we don’t know what OX should be. In a t-test we are forced to estimate it from the sample standard deviation, sX. This means that for a t-test the standard error isn’t solely a mental construct but also depends on an attribute of the sample. This is an exception to the rule, however.
