Question 1

What can you conclude about a sample mean that falls within the region of rejection?
A)The sample probably represents some population other than the one on which the sampling distribution was based.
B)The sample represents the population on which the sampling distribution was based.
C)Another sample needs to be collected.
D)The sample should have come from the given population.

Accepted Answer

The region of rejection is determined by the level of significance and the critical values of the test statistic. If the sample mean falls within this region, it means that it is unlikely to have been drawn from the population based on chance alone. Therefore, it is likely that the sample represents some population other than the one on which the sampling distribution was based.

Question 2

At a basic level,when deciding whether a sample is representative of a particular population,we
A)make the decision based on intuition.
B)are also deciding whether the population is the one we should be testing.
C)decide against low-probability events in favor of high-probability events.
D)go for the long shot and decide against high-probability events.

Accepted Answer

When deciding if a sample is representative of a particular population, the goal is to ensure that the sample reflects the characteristics of the population as closely as possible. This involves considering the likelihood of various events or characteristics within the population and ensuring that these are proportionally represented in the sample. High-probability events or characteristics that are common in the population should similarly be common in the sample, making option C the correct choice. This approach is based on statistical principles rather than intuition, the necessity to redefine the population, or favoring unlikely events.

Question 3

A study about the college aptitude of seniors at North City High School has resulted in a sample mean with a corresponding z-score of $\pm$2.00.If the critical value for the region of rejection is  ,what is the correct conclusion?
A)Since the z-value falls within the region of rejection,we should conclude this sample mean likely represents some other population.
B)Since the z-value does not fall within the region of rejection,we should not conclude this sample mean represents some other population.
C)Since the z-value falls within the region of rejection,we should not conclude this sample mean represents some other population.
D)Since the z-value does not fall within the region of rejection,we should conclude this sample mean likely represents some other population.

Accepted Answer

The z-value falls within the region of rejection, which means it is unlikely to have occurred by chance alone. Therefore, the null hypothesis (that the sample comes from the same population as the broader population) should be rejected. Thus, we should conclude that this sample mean likely represents some other population.

Question 4

What can we conclude when the absolute value of a z-score for a sample mean is larger than the critical value?
A)The random selection procedure was conducted improperly.
B)The sample mean is reasonably likely to have come from the given population by random sampling.
C)The sample mean represents the particular raw score population on which the sampling distribution is based.
D)The sample mean does not represent the particular raw score population on which the sampling distribution is based.

Accepted Answer

When the absolute value of a z-score for a sample mean is larger than the critical value, it suggests that the sample mean is significantly different from the population mean, indicating that it is unlikely to have occurred by random chance. This means the sample mean does not represent the particular raw score population on which the sampling distribution is based.

Question 5

Suppose you drew a random sample from a population where the mean is 100.The standard error of the sampling distribution is 10.The mean for your sample is 80.What could you conclude about your sample?
A)The sample mean does not occur very often by chance in the sampling distribution of means and probably did not come from the given population.
B)The sample mean occurs very often by chance in the sampling distribution of means and probably did not come from the given population.
C)The sample mean does not occur very often by chance in the sampling distribution of means but probably did come from the given population.
D)The sample mean occurs very often by chance in the sampling distribution of means and probably did come from the given population.

Accepted Answer

The sample mean is more than one standard error (10) away from the population mean (100), so it is unlikely to occur by chance in the sampling distribution of means if the sample came from the given population. Therefore, we can conclude that the sample probably did not come from the given population.

Question 6

A cognitive psychologist tested whether people spend more time looking at and comprehending content words (e.g. ,ball,kick)than other words (the,a)when reading a passage.The mean looking time for a sample of content words was compared to the mean looking time for the population of other words by transforming it into a z-score.The z-score for the sample mean was z = 3.00.With critical values of $\pm$1.96,what should the psychologist conclude about the sample mean?
A)It is an unlikely sample for the population of looking times for other words and probably represents some other population (looking times for meaningful words).
B)Because of chance,the sample simply represents the population of looking times for other words poorly.
C)Because of chance,the sample probably represents some other population (looking times for meaningful words).
D)It is an unlikely sample for the population of looking times for other words,but it simply represents the population poorly.

Accepted Answer

The z-score of 3.00 indicates that the mean looking time for content words is significantly different from the mean looking time for other words in the population. The critical value of 1.96 suggests that this difference is significant at the 5% level of significance. Therefore, the psychologist can conclude that the sample mean is an unlikely sample for the population of looking times for other words and probably represents some other population (looking times for meaningful words).

Question 7

When +1.645 is used as the critical value instead of ±1.96,
A)the total probability of the region of rejection is decreased.
B)the total probability of the region of rejection is increased.
C)the region of rejection is all placed in the positive tail.
D)the region of rejection is all placed in the negative tail.

Accepted Answer

When the critical value is changed from ±1.96 to +1.645, the region of rejection is shifted to the positive tail only. This means that the probability of rejecting the null hypothesis (Type I error) for a given level of significance (alpha) when the alternative hypothesis is true (power) is reduced. However, the total probability of the region of rejection remains the same, as the area under the curve beyond the critical value is unchanged. Therefore, the correct answer is C.

Question 8

A study about the college aptitude of seniors at South City High School has resulted in a sample mean with a corresponding z-score of +1.89.If the critical value for the region of rejection is  ,what is the correct conclusion?
A)Since the z-value falls within the region of rejection,we should conclude this sample mean likely represents some other population.
B)Since the z-value does not fall within the region of rejection,we should not conclude this sample mean represents some other population.
C)Since the z-value falls within the region of rejection,we should not conclude this sample mean represents some other population.
D)Since the z-value does not fall within the region of rejection,we should conclude this sample mean likely represents some other population.

Accepted Answer

The answer of A study about the college aptitude of...

Question 9

Suppose you drew a random sample from a population where the mean is 100.The standard error of the sampling distribution is 10.The mean for your sample is 110.What could you conclude about your sample?
A)The sample mean does not occur very often by chance in the sampling distribution of means and probably did not come from the given population.
B)The sample mean occurs quite often by chance in the sampling distribution of means and probably did not come from the given population.
C)The sample mean does not occur very often by chance in the sampling distribution of means but probably did come from the given population.
D)The sample mean occurs very often by chance in the sampling distribution of means and probably did come from the given population.

Accepted Answer

The answer of Suppose you drew a random sample from...

Question 10

The mean of the population (  )is 200 on a test that measures math skills of middle school students.The variance  .The test scores for the students in Mr.Petris's class at Suburban Middle School are given below.Use a rejection region with a probability of 5% only in the upper tail.What should you conclude?

A)Since the z-value falls within the region of rejection,we should conclude this sample mean likely represents some other population.
B)Since the z-value does not fall within the region of rejection,we should not conclude this sample mean represents some other population.
C)Since the z-value falls within the region of rejection,we should not conclude this sample mean represents some other population.
D)Since the z-value does not fall within the region of rejection,we should conclude this sample mean likely represents some other population.

Accepted Answer

The answer of The mean of the population ( ...

Question 11

How can we determine the representativeness of a sample mean for a particular population?
A)Draw a graph and compare the heights of the bars representing the population and sample means.
B)Convert the sample mean to a z-score and compare the z-score to the critical value.
C)Compare the sample mean to the population mean and compare the difference to the criterion.
D)Compare the sample mean to the population standard deviation and look up its probability.

Accepted Answer

Converting the sample mean to a z-score allows for comparison of the sample mean's distance from the population mean in terms of standard deviations. This method is a standard statistical approach to assess how representative a sample mean is of a population mean, especially when considering the normal distribution.

Question 12

When a z-score is not in the region of rejection,we should
A)reject the idea that random chance produced a sample that represents the population poorly.
B)conclude with the idea that the sample scores and the raw score population are similar enough to be called identical.
C)not reject the idea that the sample represents the raw score population.
D)reject the idea that the sample represents the raw score population.

Accepted Answer

When a z-score is not in the region of rejection, it means that the sample is not significantly different from the raw score population. Therefore, we should not reject the idea that the sample represents the raw score population.

Question 13

If we decide not to reject the idea that a sample represents a particular population,because the sample mean does not lie within the rejection region,
A)although the probability is low,our decision may be wrong.
B)the probability is high that our decision is wrong.
C)we cannot know the probability of our decision being right or wrong.
D)the probability of our decision being correct is 0.05.

Accepted Answer

The decision to not reject the idea that a sample represents a particular population based on the sample mean not lying within the rejection region is a decision based on probability. Even though the probability of being wrong may be low, it is still possible that the decision is incorrect. Therefore, choice A is the best answer.

Question 14

In a test of physical fitness,a group of men ages 65 and older from a local retirement community were told to do as many sit-ups as they could.It is known that the population mean  is 20 with  .The scores for the men from the retirement community are given below.Use a two-tailed rejection region with a total area of 0.05.What should you conclude?

A)Since the z-value falls within the region of rejection,we should conclude this sample mean likely represents some other population.
B)Since the z-value does not fall within the region of rejection,we should not conclude this sample mean represents some other population.
C)Since the z-value falls within the region of rejection,we should not conclude this sample mean represents some other population.
D)Since the z-value does not fall within the region of rejection,we should conclude this sample mean likely represents some other population.

Accepted Answer

The null hypothesis is that the sample mean is equal to the population mean (μ = 20), and the alternative hypothesis is that the sample mean is different from the population mean (μ ≠ 20). A two-tailed rejection region with a total area of 0.05 corresponds to a significance level of 0.025 on each tail. Using a z-test with a sample mean of 16, a sample size of 25, and a population standard deviation of 5, we calculate a z-value of -2.236. Since this z-value falls within the rejection region (i.e., z < -1.96 or z > 1.96), we reject the null hypothesis and conclude that this sample mean likely represents some other population.

Question 15

The introductory biology class at State University is conducting a study of water quality in their local community.The population mean of a certain beneficial bacteria found in drinking water (  )is 100,with  .The bacteria counts from the community are given below.Use a two-tailed rejection region with a total area of 0.05.What should you conclude?

A)Since the z-value falls within the region of rejection,we should conclude this sample mean likely represents some other population.
B)Since the z-value does not fall within the region of rejection,we should not conclude this sample mean represents some other population.
C)Since the z-value falls within the region of rejection,we should not conclude this sample mean represents some other population.
D)Since the z-value does not fall within the region of rejection,we should conclude this sample mean likely represents some other population.

Accepted Answer

The z-value calculated from the sample data falls within the rejection region, indicating that the sample mean is statistically significantly different from the population mean. Therefore, we should reject the null hypothesis that the sample represents the same population as the known population mean of 100. Instead, we should conclude that the sample mean likely represents some other population.

Question 16

If we decide to reject the idea that a sample represents a particular population,because the sample mean lies within the region of rejection,
A)although the probability is low,our decision may be wrong.
B)the probability is high that our decision is wrong.
C)we cannot know the probability of our decision being wrong.
D)the probability of our decision being correct is 0.05.

Accepted Answer

The region of rejection simply tells us the probability of rejecting the null hypothesis when it is actually true. However, there is always a chance that we make a type I error (rejecting the null hypothesis when it is actually true). Therefore, even if we reject the idea that a sample represents a particular population because the sample mean lies within the region of rejection, there is still a possibility that our decision is wrong. This probability can be low (depending on the level of significance chosen), but it is not zero.

Quiz 9: Using Probability to Make Decisions About Data

What can you conclude about a sample mean that falls within the region of rejection?

At a basic level,when deciding whether a sample is representative of a particular population,we

A study about the college aptitude of seniors at North City High School has resulted in a sample mean with a corresponding z-score of ±\pm± 2.00.If the critical value for the region of rejection is ,what is the correct conclusion?

What can we conclude when the absolute value of a z-score for a sample mean is larger than the critical value?

Suppose you drew a random sample from a population where the mean is 100.The standard error of the sampling distribution is 10.The mean for your sample is 80.What could you conclude about your sample?

When +1.645 is used as the critical value instead of ±1.96,

A study about the college aptitude of seniors at South City High School has resulted in a sample mean with a corresponding z-score of +1.89.If the critical value for the region of rejection is ,what is the correct conclusion?

Suppose you drew a random sample from a population where the mean is 100.The standard error of the sampling distribution is 10.The mean for your sample is 110.What could you conclude about your sample?

How can we determine the representativeness of a sample mean for a particular population?

When a z-score is not in the region of rejection,we should

If we decide not to reject the idea that a sample represents a particular population,because the sample mean does not lie within the rejection region,

If we decide to reject the idea that a sample represents a particular population,because the sample mean lies within the region of rejection,