Questions and Answers
What is the mean of the sampling distribution of the difference of the sample means?
If we are trying to establish that the mean of population 1 is greater than the mean of population 2, what is the appropriate set of hypotheses?
If we are trying to establish that the mean of population 1 is not the same as the mean of population 2, what is the appropriate set of hypotheses?
What is the point estimate for the difference between the two population means (µA - µB)?
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What is the standard deviation (standard error) for the distribution of differences of sample means (xA - xB)?
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What is the computed test statistic for determining whether the average amount dispensed by machine A is significantly more than that by machine B?
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What is the P value for the test to determine if the average amount dispensed by machine A is significantly more than that by machine B?
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What is the point estimate for p if we assume that the two population proportions are equal?
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What are the appropriate hypotheses if we are testing if remediation in mathematics has a positive effect on students?
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What is the computed test statistic for the appropriate test regarding the effects of remediation in mathematics?
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A claim or statement about a population parameter is classified as the null hypothesis.
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A statement contradicting the claim in the null hypothesis about a population parameter is classified as the alternative hypothesis.
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If we want to claim that a population parameter is different from a specified value, this situation can be considered as a one-tailed test.
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The null hypothesis is considered correct until proven otherwise.
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A Type I error is the error we make when we fail to reject an incorrect null hypothesis.
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The probability of making a Type I error and the level of significance are equal or the same.
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The range of z values that indicates there is a significant difference between the value of the sample statistic and the proposed parameter value is called the rejection region or the critical region.
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If the sample size n is less than 30, then a z score will always be associated with any hypothesis that deals with the mean.
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In the P-value approach to hypothesis testing, if the P value is less than a specified significance level, we fail to reject the null hypothesis.
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In the P-value approach to hypothesis testing, if 0.01 < P value < 0.05, there is insufficient evidence to reject the null hypothesis.
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In the P-value approach to hypothesis testing, if P value < 0.001, there is very strong evidence to reject the null hypothesis.
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When large samples (n≥30) are associated with hypothesis tests for population proportions, the associated test statistic is a z score.
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The distribution of sample proportions from a single population is approximately normal provided that the sample size is large enough (n≥30).
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The distribution of the difference between two sample means is approximately normal with variance + , where n1 and n2 are the sample sizes from populations 1 and 2, respectively, and σ1^2 and σ2^2 are the respective variances, if the sample sizes are both greater than or equal to 30.
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In testing for the difference of two population means, if the population variances are unknown and the sample sizes from the populations are both greater than or equal to 30, the associated test statistic is approximately a z score.
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If the null hypothesis is rejected, this means that the null hypothesis is not true.
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When performing hypothesis tests on two population means, it is necessary to assume that the populations are normally distributed.
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The P value of a hypothesis test can be computed without the value of the test statistic.
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The P value of a hypothesis test is the smallest level of significance at which the null hypothesis can be rejected.
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In hypothesis testing, the alternative hypothesis is assumed to be true.
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In hypothesis testing, if the null hypothesis is rejected, the alternative hypothesis must also be rejected.
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The calculated numerical value that is compared to a table value in a hypothesis test is called the
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A right-tailed test is conducted with α = 0.0582. If the z tables are used, the critical value will be
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A right-tailed test is performed, with the test statistic having a standard normal distribution. If the computed test statistic is 3.00, the P value for this test is
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New software is being integrated into the teaching of a course with the hope that it will help to improve the overall average score for this course. The historical average score for this course is 72. If a statistical test is done for this situation, the alternative hypothesis will be
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Dr. J claims that 40 percent of his College Algebra class (a very large section) will drop his course by midterm. To test his claim, he selected 45 names at random and discovered that 20 of them had already dropped long before midterm. The test statistic value for his hypothesis test is
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In testing a hypothesis, the hypothesis that is assumed to be true is
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A Type I error is defined to be the probability of
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A Type II error is defined to be the probability of
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In hypothesis testing, the level of significance is the probability of
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The level of significance can be any
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If you fail to reject the null hypothesis in the testing of a hypothesis, then
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If we were testing the hypotheses H0: µ = µ0 vs. H1: µ > µ0 at a given significance level α, with large samples and unknown population variance, then H0 will be rejected if the computed test statistic is
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Which of the following general guidelines is used when using the P value to perform hypothesis tests?
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When the P value is used in testing a hypothesis, we will not reject the null hypothesis for a level of significance α when
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A real estate agent claims that the average price for homes in a certain subdivision is $150,000. You believe that the average price is lower. If you plan to test his claim by taking a random sample of the prices of the homes in the subdivision, the formulated set of hypotheses will be
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A statistics student was not pleased with his final grade in his statistics course, so he decided to appeal his grade. He believes that the average score on the final examination was less than 69 (out of a possible 100 points), so he believes that it was an unfair examination. He thinks that he should have made at least a grade of B in the course. He decided to test his claim about the average of the final examination. If he knows his 'statistics,' the correct set of hypothesis he will set up to test his claim is
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An advertisement on the TV claims that a certain brand of tire has an average lifetime of 50,000 miles. Suppose you plan to test this claim by taking a sample of tires and putting them on test. The correct set of hypotheses to set up is
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The local newspaper reported that at least 25 percent of the population in a university community works at the university. You believe that the proportion is lower. If you selected a random sample to test this claim, the appropriate set of hypotheses would be
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The local newspaper claims that 15 percent of the residents of the community play the state lottery. If you plan to test the claim by taking a random sample from the community, the appropriate set of hypotheses is
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The local newspaper claims that no more than 5 percent of the residents of the community are on welfare. If you plan to test the claim by taking a random sample from the community, the appropriate set of hypotheses is
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For the following information, n = 16, µ = 15, x = 16, σ^2 = 16; assume that the population is normal and compute the test statistic if you were testing for a single population mean.
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For the following information, n = 16, µ = 15, x = 16, σ^2 = 16; assume that the population is normal. If you are performing a right-tailed test for a single population mean, then
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For the following information, n = 16, µ = 15, x = 16, σ^2 = 16; assume that the population is normal. If you are performing a right-tailed test for a single population mean, then you
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For the following information, n = 16, µ = 15, x = 16, σ^2 = 16; assume that the population is normal. If you are performing a left-tailed test for a single population mean, then you
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If a null hypothesis is rejected at the 0.05 level of significance for a two-tailed test, you
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If a null hypothesis is rejected at the 5 percent significance level for a right-tailed test, you
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For a left-tailed test concerning the population proportion with sample size 203 and α = 0.05, the null hypothesis will be rejected if the computed test statistic is
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It was reported that a certain population had a mean of 27. To test this claim, you selected a random sample of size 100. The computed sample mean and sample standard deviation were 25 and 7, respectively. The appropriate set of hypotheses for this test is
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It was reported that a certain population had a mean of 27. To test this claim, you selected a random sample of size 100. The computed sample mean and sample standard deviation were 25 and 7, respectively. The computed test statistic for the appropriate set of hypotheses is
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It was reported that a certain population had a mean of 27. To test this claim, you selected a random sample of size 100. The computed sample mean and sample standard deviation were 25 and 7, respectively. The P value for the appropriate set of hypotheses is
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It was reported that a certain population had a mean of 27. To test this claim, you selected a random sample of size 100. The computed sample mean and sample standard deviation were 25 and 7, respectively. At the 0.05 level of significance, you can claim that the average of this population is
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For a highly publicized murder trial, it was estimated that 25 percent of the population watched the proceedings on TV. A statistics student felt that this estimate was too small for his community and decided to do a hypothesis test. He selected a random sample of 100 residents from the university community where he lives and found that 32 of them actually watched at least three hours of the proceedings. The appropriate set of hypotheses for the test is
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For a highly publicized murder trial, it was estimated that 25 percent of the population watched the proceedings on TV. A statistics student felt that this estimate was too small for his community and decided to do a hypothesis test. He selected a random sample of 100 residents from the university community where he lives and found that 32 of them actually watched at least three hours of the proceedings. The computed test statistic for the test is
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For a highly publicized murder trial, it was estimated that 25 percent of the population watched the proceedings on TV. A statistics student felt that this estimate was too small for his community and decided to do a hypothesis test. He selected a random sample of 100 residents from the university community where he lives and found that 32 of them actually watched at least three hours of the proceedings. The P value for the test is
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For a highly publicized murder trial, it was estimated that 25 percent of the population watched the proceedings on TV. A statistics student felt that this estimate was too small for his community and decided to do a hypothesis test. He selected a random sample of 100 residents from the university community where he lives and found that 32 of them actually watched at least three hours of the proceedings. At the 10 percent significance level, you can claim that the proportion of viewers in this community was
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For a highly publicized murder trial, it was estimated that 25 percent of the population watched the proceedings on TV. A statistics student felt that this estimate was too small for his community and decided to do a hypothesis test. He selected a random sample of 100 residents from the university community where he lives and found that 32 of them actually watched at least three hours of the proceedings. The standard deviation for the distribution of the sample proportion is
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Study Notes
Hypothesis Testing Concepts
- A null hypothesis (H0) asserts a statement about a population parameter, presumed true until proven otherwise.
- An alternative hypothesis (H1) contradicts the null hypothesis, suggesting a different parameter value.
- A one-tailed test evaluates if a parameter is significantly greater or less than a specified value, while a two-tailed test assesses if it is simply different.
Errors in Hypothesis Testing
- A Type I error occurs when a true null hypothesis is incorrectly rejected.
- A Type II error happens when a false null hypothesis is not rejected.
- Significance level (α) represents the probability of making a Type I error; it is equal to the probability of rejection when the null hypothesis is true.
Test Statistics and Regions
- The rejection region is determined by critical values of the test statistic (often z-scores), which indicates a significant difference from a proposed parameter value.
- For large sample sizes (n ≥ 30), the test statistic often follows a normal distribution (z-distribution) regardless of the population variance.
- Sample proportions from large populations also follow an approximately normal distribution with sufficient sample size.
P-Value in Hypothesis Testing
- The p-value indicates the strength of evidence against the null hypothesis; a p-value less than α suggests rejecting H0.
- If the p-value is less than 0.001, it provides very strong evidence to reject H0; if between 0.01 and 0.05, there is moderate evidence to reject H0.
Setting Up Hypotheses
- When testing claims, set hypotheses based on the claim: for instance, if claiming a population mean is lower than a known mean, set H0 as the population mean being equal or higher.
- Specific examples include:
- Testing average home prices or population parameters involves formulating H0 and H1 correctly.
- For a population mean of 27, if the sample mean is 25, the hypotheses setup is H0: µ = 27 vs. H1: µ ≠ 27.
Critical Values and Test Results
- Critical values are determined based on the significance level and nature of the test (one-tailed vs. two-tailed).
- Decisions to reject or fail to reject the null depend on comparing test statistics against critical values.
- For instance, in a right-tailed test, reject H0 if test statistic exceeds the critical value defined according to α.
Summary of Calculated Values
- Test statistics and corresponding p-values should be calculated and interpreted based on sample data and size.
- Results from tests done with different hypotheses often provide various insights and necessitate a careful examination of evidence to make conclusions about population parameters.
Practical Applications
- Hypothesis testing applies across diverse fields, including education, health, and commerce for assessing claims backed by statistics.
- Understanding the statistics behind different hypothesis tests enables better decision-making in both academic and professional settings.### Machine A and Machine B - Dog Food Filling
- Sample Sizes: Machine A has 81 samples; Machine B has 64 samples.
- Sample Means: Machine A dispenses an average of 51 lbs; Machine B dispenses 48 lbs.
- Sample Variances: Machine A has a variance of 16; Machine B has a variance of 12.
- Objective: Test if the average dispensed by Machine A is significantly greater than that of Machine B.
Hypothesis Testing
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Hypotheses:
- Null Hypothesis (H0): µA - µB ≤ 0
- Alternative Hypothesis (H1): µA - µB > 0 (indicates that Machine A dispenses more).
Test Statistics
- Test Statistic Computation: Calculated value needed to assess the null hypothesis.
- Possible Outcomes: Computed test statistic options include approximately 7.7918 and 2.1988.
P-Value
- P-Value Significance: Indicates the probability of observing the test results under the null hypothesis.
- Potential P-Value Ranges: Options include approximately 0.5, 0.0, and 1.0.
Remedial vs Nonremedial Study
- Sample Sizes: Remedial group (N1) has 34; Nonremedial group (N2) has 150.
- Success Rates: Remedial group has 20 successes; Nonremedial group has 104 successes.
Proportion Estimation
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Point Estimate for Population Proportion (p):
- Calculated from successes and total sample sizes.
- Different options for point estimates include approximately 0.4565, 0.6739, and 0.4078.
Standard Deviation
- Estimate for Standard Deviation of Differences: Measures variability in sample proportions.
- Calculated Values: Options for standard deviation estimations include approximately 0.0079, 0.1898, and 0.0360.
Hypotheses for Remediation Study
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Proportions of Success:
- Null Hypothesis (H0): pr - pn ≤ 0 (indicates no effect of remediation).
- Alternative Hypothesis (H1): pr - pn > 0 (indicates a positive effect of remediation).
Test Statistic for Success Rates
- Computed Test Statistic: Comparisons between the two groups based on the number of successes.
- Options for Test Stat Values: Include values such as 3.6426 and -1.1803.
Studying That Suits You
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Description
Test your knowledge of key concepts from Chapter 12 of Statistics. This set of flashcards focuses on hypotheses, including null and alternative hypotheses, as well as testing methods. Perfect for anyone studying statistics and looking to reinforce their understanding of these fundamental concepts.
