Significance Logic and Establishing Hypotheses - AP Statistics
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A pretzel company advertises that their pretzels contain less than 1.0g of of sodium per serving. You take a simple random sample of 10 pretzel servings, and calculate that the mean amount of sodium is 1.20 g, with a standard deviation of 0.1 g.
At the 95% confidence level, does your sample suggest that the pretzels actually have higher than 1.0g of sodium per serving?
A pretzel company advertises that their pretzels contain less than 1.0g of of sodium per serving. You take a simple random sample of 10 pretzel servings, and calculate that the mean amount of sodium is 1.20 g, with a standard deviation of 0.1 g.
At the 95% confidence level, does your sample suggest that the pretzels actually have higher than 1.0g of sodium per serving?
This is a one- tailed t-test. It is one-tailed because the question asks whether the pretzel's mean is actually higher, so we are only interested in the right hand tail. We will be using the t-distribution because the population standard deviation is not known.
First we write our hypotheses:
Now we need the appropriate formula for a t-test. We will be using standard error because we are working with the standard deviation of a sampling distribution.
Now we fill in the values from our problem
Now we must look up the t-critical value, or use technology to find the p-value.
We must find the t-critcal value by finding 
Because our test statistic 6.32 is more extreme than our critical value, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0.
If you calculated a p-value using technology, p=0.00006884.
Because 
, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0 g.
This is a one- tailed t-test. It is one-tailed because the question asks whether the pretzel's mean is actually higher, so we are only interested in the right hand tail. We will be using the t-distribution because the population standard deviation is not known.
First we write our hypotheses:
Now we need the appropriate formula for a t-test. We will be using standard error because we are working with the standard deviation of a sampling distribution.
Now we fill in the values from our problem
Now we must look up the t-critical value, or use technology to find the p-value.
We must find the t-critcal value by finding
Because our test statistic 6.32 is more extreme than our critical value, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0.
If you calculated a p-value using technology, p=0.00006884.
Because
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James goes to UCLA, and he believes that the atheletes of UCLA are better runners, than the country average. He did a bit of a research and found that the national average time for a two-mile run for college atheletes is 
min with a standard deviation of 
minute. He then sampled 
UCLA atheletes and found that their average two-mile time was 
minutes.
Is James' data statistically significant? Can we confirm that UCLA atheletes are better than average runners? And if so, to which level of certainty: 
, 
, 
, 
James goes to UCLA, and he believes that the atheletes of UCLA are better runners, than the country average. He did a bit of a research and found that the national average time for a two-mile run for college atheletes is
Is James' data statistically significant? Can we confirm that UCLA atheletes are better than average runners? And if so, to which level of certainty:
Using a Z-test (we have population SD, not sample SD) and a population of 
, we arrive at a P-value of 
, which is lower than 
, but above 
.
Using a Z-test (we have population SD, not sample SD) and a population of
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Bob and Alvin suspect that the weight of the average man in Jackson, Mississippi is significantly different than the weight of the average man in Green Bay, Wisconsin. They sample 100 men in Jackson and find that the average weight is 191 lbs, and the sample standard deviation is 30 lbs.
They sample 100 men in Green Bay and find that the average weight is 184, and the sample standard deviation is 25.
Bob says that they should present a 95% 1-tailed test where the alternate hypothesis is:
and the null hypothesis is:
Alvin disagrees; he says that a one-tailed test assumes that we already suspected a higher weight in Jackson. He recommends a 95% 2-tailed test where
and
.
Show why the 1-tailed test rejects its null hypothesis and the 2-tailed test fails to reject its null hypothesis. Provide the following:
-
The Z-value for a 1-tailed 95% test.
-
The Z-value for a 2-tailed 95% test.
-
The Z-value for the sample difference of 7.
Bob and Alvin suspect that the weight of the average man in Jackson, Mississippi is significantly different than the weight of the average man in Green Bay, Wisconsin. They sample 100 men in Jackson and find that the average weight is 191 lbs, and the sample standard deviation is 30 lbs.
They sample 100 men in Green Bay and find that the average weight is 184, and the sample standard deviation is 25.
Bob says that they should present a 95% 1-tailed test where the alternate hypothesis is:
and the null hypothesis is:
Alvin disagrees; he says that a one-tailed test assumes that we already suspected a higher weight in Jackson. He recommends a 95% 2-tailed test where
and
Show why the 1-tailed test rejects its null hypothesis and the 2-tailed test fails to reject its null hypothesis. Provide the following:
-
The Z-value for a 1-tailed 95% test.
-
The Z-value for a 2-tailed 95% test.
-
The Z-value for the sample difference of 7.
The variance for the sample difference is:
The standard deviation for the sample difference is:
The Z-value for a difference of 7 is:
The sample difference Z-value is greater than the 1-tailed Z-value (causing us to reject the null hypothesis) and is less than the 2-tailed Z-value (causing us to fail to reject the null hypothesis).
The variance for the sample difference is:
The standard deviation for the sample difference is:
The Z-value for a difference of 7 is:
The sample difference Z-value is greater than the 1-tailed Z-value (causing us to reject the null hypothesis) and is less than the 2-tailed Z-value (causing us to fail to reject the null hypothesis).
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Jimmy thinks that Josh cannot shoot more than 50 points on average in a game. Josh disputes this claim and tells Jimmy that he is going to play 10 games and prove him wrong. What is the null hypothesis?
Jimmy thinks that Josh cannot shoot more than 50 points on average in a game. Josh disputes this claim and tells Jimmy that he is going to play 10 games and prove him wrong. What is the null hypothesis?
The null hypothesis is what we intend to either reject or fail to reject using our sample data. In this case, the null hypothesis is that Josh cannot shoot more than 50 points on average, and Josh's performance in 10 games are the sample data we use to assess this hypothesis.
The null hypothesis is what we intend to either reject or fail to reject using our sample data. In this case, the null hypothesis is that Josh cannot shoot more than 50 points on average, and Josh's performance in 10 games are the sample data we use to assess this hypothesis.
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A student is beginning an analysis to determine whether there is a relationship between temperatures and traffic accidents. The student is trying to articulate a null hypothesis for the study. Which of the following is an acceptable null hypothesis?
A student is beginning an analysis to determine whether there is a relationship between temperatures and traffic accidents. The student is trying to articulate a null hypothesis for the study. Which of the following is an acceptable null hypothesis?
The null hypothesis is the default hypothesis and predicts that there is no relationship between the variables in question. Each of the incorrect answer choices here either predicts a relationship between variables or makes a broad assertion that includes much more than the variables in question.
The null hypothesis is the default hypothesis and predicts that there is no relationship between the variables in question. Each of the incorrect answer choices here either predicts a relationship between variables or makes a broad assertion that includes much more than the variables in question.
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A statistician has determined that she will reject the null hypothesis if she can have 
confidence that there is a statistically significant relationship between the variables in question. She conducts a statistical analysis and obtains a p value of 
. Should the statistician reject the null hypothesis?
A statistician has determined that she will reject the null hypothesis if she can have
The statistician has determined that she will only reject the null hypothesis if she has 95% confidence that there is a relationship between variables.
To have this level of confidence, the statistician must obtain a p value of 0.05 or lower.
Therefore, she should not reject the null hypothesis since 0.1 is greater that 0.05.
The statistician has determined that she will only reject the null hypothesis if she has 95% confidence that there is a relationship between variables.
To have this level of confidence, the statistician must obtain a p value of 0.05 or lower.
Therefore, she should not reject the null hypothesis since 0.1 is greater that 0.05.
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The Environmental Protection Agency (EPA) wants to test the pollution level of the Colorado River. If the pollution level is too high, the water will be stopped from going into drinking water pipelines. The EPA randomly chooses different spots along the river to collect water samples from, and then tests the samples for their pollution levels. Which of the following decisions would result from the type I error?
The Environmental Protection Agency (EPA) wants to test the pollution level of the Colorado River. If the pollution level is too high, the water will be stopped from going into drinking water pipelines. The EPA randomly chooses different spots along the river to collect water samples from, and then tests the samples for their pollution levels. Which of the following decisions would result from the type I error?
The hypotheses tested here are:
The true mean pollution level is within the allowed limit.
The true mean pollution level is higher than the allowed limit.
The type I error occurs when the null hypothesis is rejected even though it is actually true. In this case, the type I error would be deciding that the mean pollution levels are higher than the allowed limit and closing the drinking water pipelines.
The hypotheses tested here are:
The type I error occurs when the null hypothesis is rejected even though it is actually true. In this case, the type I error would be deciding that the mean pollution levels are higher than the allowed limit and closing the drinking water pipelines.
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A study would like to determine whether meditation helps students improve focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate null hypothesis for this study?
A study would like to determine whether meditation helps students improve focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate null hypothesis for this study?
Because we are comparing two samples, the hypothesis takes the form of Mu1- Mu2. Because we are testing the claim that meditation increases average study time, the null hypothesis must cover all other outcomes. That means the null hypothesis is that the difference between control minus meditation is greater than or equal to zero.
Because we are comparing two samples, the hypothesis takes the form of Mu1- Mu2. Because we are testing the claim that meditation increases average study time, the null hypothesis must cover all other outcomes. That means the null hypothesis is that the difference between control minus meditation is greater than or equal to zero.
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A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate null hypothesis for this study?
A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate null hypothesis for this study?
Because we are comparing one sample to a known mean, the hypothesis is 
compared to the known average, 23. We would like to test the claim that meditating will increase focus time, therefore our null hypothesis must include all other outcomes. Therefore, the null is that Mu is less than or equal to 23.
Because we are comparing one sample to a known mean, the hypothesis is
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A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate null hypothesis for this study?
A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate null hypothesis for this study?
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. Therefore, the null hypothesis is that the relationship is not significant, meaning the slope of the line is equal to zero.
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. Therefore, the null hypothesis is that the relationship is not significant, meaning the slope of the line is equal to zero.
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A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct null hypothesis?
A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct null hypothesis?
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that all of the groups’ means are the same.
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that all of the groups’ means are the same.
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A researcher wants to investigate the claim that taking vitamins will help a student study longer. First, the researcher collects 32 students who do not take vitamins and determines their time spent studying. Then, the 32 students are given a vitamin for 1 week. After 1 week of taking vitamins, students are again tested to determine their time spent studying. Which of the following is the correct null hypothesis?
A researcher wants to investigate the claim that taking vitamins will help a student study longer. First, the researcher collects 32 students who do not take vitamins and determines their time spent studying. Then, the 32 students are given a vitamin for 1 week. After 1 week of taking vitamins, students are again tested to determine their time spent studying. Which of the following is the correct null hypothesis?
Because the same students are tested twice, this is a paired study, therefore we must use a hypothesis appropriate for a paired t-test. The hypothesis for a paired t-test regards the average of the differences between before and after treatment, called MuD. We are testing the claim that vitamins increase study time, which would mean that study time for vitamin users would be greater than that of the control. Therefore the null must include all other outcomes. The null hypothesis should state that the difference between before and after treatment is greater than or equal to zero.
Because the same students are tested twice, this is a paired study, therefore we must use a hypothesis appropriate for a paired t-test. The hypothesis for a paired t-test regards the average of the differences between before and after treatment, called MuD. We are testing the claim that vitamins increase study time, which would mean that study time for vitamin users would be greater than that of the control. Therefore the null must include all other outcomes. The null hypothesis should state that the difference between before and after treatment is greater than or equal to zero.
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For her school science project, Susy wants to determine whether the ants in her neighborhood have smaller colonies than average. Research tells her that the average Harvester colony has around 4,000 ants. She counts the number of ants in 5 colonies in her neighborhood and determines the average colony size to be 3,700 ants. What is the appropriate null hypothesis for her science project?
For her school science project, Susy wants to determine whether the ants in her neighborhood have smaller colonies than average. Research tells her that the average Harvester colony has around 4,000 ants. She counts the number of ants in 5 colonies in her neighborhood and determines the average colony size to be 3,700 ants. What is the appropriate null hypothesis for her science project?
Susy wants to know whether ants in her neighborhood have smaller colonies, so that will be her alternative hypothesis. Therefore her null hypothesis needs to cover all other outcomes, that the colony sizes are greater than or equal to average colony size of 4000 ants.
Susy wants to know whether ants in her neighborhood have smaller colonies, so that will be her alternative hypothesis. Therefore her null hypothesis needs to cover all other outcomes, that the colony sizes are greater than or equal to average colony size of 4000 ants.
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For his school science project, Timmy wants to determine whether the ants in his neighborhood have colonies that are sized differently than normal. His research shows that the average Harvester colony has around 4000 ants. He counts the number of ants in 5 colonies and determines that the average colony size is 3,700 ants. What is the appropriate null hypothesis for his science project?
For his school science project, Timmy wants to determine whether the ants in his neighborhood have colonies that are sized differently than normal. His research shows that the average Harvester colony has around 4000 ants. He counts the number of ants in 5 colonies and determines that the average colony size is 3,700 ants. What is the appropriate null hypothesis for his science project?
Timmy does not have a directional hypothesis, he only wants to know whether local ant colonies are different from average. Therefore he thinks the colonies could be bigger or smaller than average. This means his alternative hypothesis is that the ant colonies are NOT equal to the average colony size of 4000 ants. His null hypothesis must include all other outcomes, which in this case is that local ant colonies are equal to the average size of 4000 ants.
Timmy does not have a directional hypothesis, he only wants to know whether local ant colonies are different from average. Therefore he thinks the colonies could be bigger or smaller than average. This means his alternative hypothesis is that the ant colonies are NOT equal to the average colony size of 4000 ants. His null hypothesis must include all other outcomes, which in this case is that local ant colonies are equal to the average size of 4000 ants.
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A researcher stated that 
percent of Americans in the 2008 presidential election were liberal. The researcher wants to find evidence that the percent of liberal Americans was different for the 2012 presidential election. Which of the hypotheses best describes the question that the researcher is trying to answer?
A researcher stated that
The question states that the researcher thinks the percent of liberal Americans in 2012 is different than the 
percent in 2008. The hypothesis is 2 directional. Therefore the null hypothesis is 
and the alternative is 
.
The question states that the researcher thinks the percent of liberal Americans in 2012 is different than the
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The department of labor reports that umemployment is down to 5% in the United States. A politician claims that the data was manipulated to show better numbers. The politician believes our country is worse off that unemployment must be worse than the claim made by the department of labor. What is the alternative hypothesis for a test for this situation?
The department of labor reports that umemployment is down to 5% in the United States. A politician claims that the data was manipulated to show better numbers. The politician believes our country is worse off that unemployment must be worse than the claim made by the department of labor. What is the alternative hypothesis for a test for this situation?
The alternative hypthesis is denoted by Ha. The alternative represents the value that is being tested for and not the original claim. The original claim for the problem is unemployment is 
%. The test claim is that it is greater than 
%. This means the alternative must be 
.
The alternative hypthesis is denoted by Ha. The alternative represents the value that is being tested for and not the original claim. The original claim for the problem is unemployment is
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A study would like to determine whether meditation helps students increase focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate alternative hypothesis for this study?
A study would like to determine whether meditation helps students increase focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate alternative hypothesis for this study?
Because we are comparing two samples, the hypothesis takes the form of 
. The study is testing the claim that meditation helps increase focus time, so this will be our alternative hypothesis. Our alternative hypothesis in words is that the difference between control and meditation groups is less than zero. That is because we think average time for meditators will be greater than the control group’s average.
Because we are comparing two samples, the hypothesis takes the form of
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A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate alternative hypothesis for this study?
A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate alternative hypothesis for this study?
Because we are comparing one sample to a known mean, the hypothesis takes the form of 
compared to the known value. We would like to test the claim that meditating will increase focus time, so this will be the alternative hypothesis. Therefore, the alternative that 
is greater than the average of 23 minutes.
Because we are comparing one sample to a known mean, the hypothesis takes the form of
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A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate alternative hypothesis for this study?
A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate alternative hypothesis for this study?
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. A slope of zero indicates no relationship between the two variables. Therefore, the alternative hypothesis is that the slope is not equal to zero.
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. A slope of zero indicates no relationship between the two variables. Therefore, the alternative hypothesis is that the slope is not equal to zero.
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A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct alternative hypothesis?
A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct alternative hypothesis?
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that not all of the groups’ means are the same. 
is not correct because this hypothesis implies that all of the groups are different, which is not actually tested by the basic ANOVA test.
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that not all of the groups’ means are the same.
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