Logo

AskSia

Plus

If we reject the null hypothesis, what can we conclude subject to the probabilit...
May 9, 2024
If we reject the null hypothesis, what can we conclude subject to the probability, α? Group of answer choices The alternative hypothesis is false. The null hypothesis is true. Reject the null with a probability, α, of making a Type I1 error. Both the null hypothesis and the alternative hypothesis are true.
Solution by Steps
step 1
Understand the concept of Type I error: Rejecting the null hypothesis when it is actually true. The probability of making a Type I error is denoted by α \alpha
step 2
Interpret the meaning of rejecting the null hypothesis at significance level α \alpha : It means that we have found sufficient evidence to support the alternative hypothesis, but there is still a α \alpha chance that we are making a Type I error
step 3
Evaluate the answer choices in the context of statistical hypothesis testing and the definition of α \alpha
Answer
C
Key Concept
Type I Error
Explanation
Rejecting the null hypothesis means there is a probability α \alpha of making a Type I error, not that the alternative hypothesis is necessarily true or that the null hypothesis is true.
◊From a Statistics perspective, a related question could be: What is the relationship between the significance level (α) and the probability of making a Type I error in hypothesis testing?⍭ Generate me a similar question◊
Multiple Choice: If we reject the null hypothesis, what can we conclude subject to the probability, α? Group of answer choices A. The alternative hypothesis is false. B. The null hypothesis is true. C. Reject the null with a probability, α, of making a Type I1 error. D. Both the null hypothesis and the alternative hypothesis are true.
Solution by Steps
step 1
Understand the concept of Type I error: Rejecting the null hypothesis when it is actually true. The probability of making a Type I error is denoted by α \alpha
step 2
Interpret the meaning of rejecting the null hypothesis at significance level α \alpha : It means that we have found sufficient evidence to support the alternative hypothesis, but there is still a probability α \alpha that this conclusion is incorrect due to a Type I error
step 3
Evaluate the answer choices in the context of statistical hypothesis testing and the definition of α \alpha
Answer
C
Key Concept
Type I Error Probability
Explanation
When we reject the null hypothesis, we conclude that there is sufficient evidence to support the alternative hypothesis, but there is a probability α \alpha that this decision is a Type I error.
Multiple Choice: A is an assumption about an unknown population parameter. Group of answer choices A. Type II error B. statistical hypothesis C. Type I error D. statistical inference
Solution by Steps
step 1
Identify the concept that relates to an assumption about an unknown population parameter
step 2
A statistical hypothesis is a statement about a population parameter that can be tested statistically
step 3
Type I and Type II errors are related to the outcomes of hypothesis testing, not the assumptions about the population parameter itself
step 4
Statistical inference involves using sample data to make generalizations about a population, but it is not an assumption itself
Answer
The correct answer is the one that directly relates to an assumption about a population parameter.
B
Key Concept
Statistical Hypothesis
Explanation
A statistical hypothesis is an assumption about a population parameter that can be tested using statistical methods.
Multiple Choice: The null hypothesis, generally referred to by H0, is the hypothesis that is tested for possible ______ under the assumption that is true. Group of answer choices A. rejection B. adjustment C. acceptance D. balance
Solution by Steps
step 1
Understand the concept of the null hypothesis, H0 H_0 , in hypothesis testing
step 2
Recognize that H0 H_0 is a statement about a population parameter that is assumed to be true for the purpose of the test
step 3
Identify that the hypothesis test is designed to assess the strength of the evidence against H0 H_0
step 4
Acknowledge that if the evidence is strong enough, we reject H0 H_0 ; otherwise, we do not reject it
step 5
Note that the term "acceptance" is not typically used because failing to reject H0 H_0 does not necessarily mean it is true, only that there is not enough evidence to conclude it is false
step 6
Understand that "adjustment" and "balance" are not relevant terms in the context of hypothesis testing for H0 H_0
Answer
A
Key Concept
Null Hypothesis Testing
Explanation
In hypothesis testing, the null hypothesis H0 H_0 is tested for possible rejection under the assumption that it is true.
Multiple Choice: The defines the smallest value of for which the null hypothesis can be rejected. Group of answer choices A: p value B: t value C: critical value D: z value
Solution by Steps
step 1
Identify the term that defines the smallest value of a statistic for which the null hypothesis can be rejected
step 2
The critical value is the point on the scale of the test statistic beyond which we reject the null hypothesis. It is determined by the significance level α \alpha , and it is the value that separates the critical region (where we reject the null hypothesis) from the values where we do not reject the null hypothesis
Answer
The correct answer is the critical value, which defines the smallest value of a statistic for which the null hypothesis can be rejected.
C
Key Concept
Critical Value
Explanation
The critical value is the threshold set by the significance level in hypothesis testing that determines the boundary for rejecting the null hypothesis.
Multiple Choice: If the critical z-value for a hypothesis test equals 2.45, what value of the test statistic would provide the least chance of making a Type I1 error? Group of answer choices A: 2.46 B: 3.74 C: 1.07 D: 4.56
Solution by Steps
step 1
Understand the concept of Type I error: A Type I error occurs when the null hypothesis is true, but we incorrectly reject it. The probability of making a Type I error is denoted by α \alpha and is determined by the critical value in a hypothesis test
step 2
Identify the critical z-value: The critical z-value for the hypothesis test is given as 2.45. This value separates the rejection region (where we would reject the null hypothesis) from the non-rejection region on the standard normal distribution curve
step 3
Determine the test statistic that minimizes Type I error: To minimize the chance of making a Type I error, we want a test statistic that falls within the non-rejection region and as far away from the critical value as possible
step 4
Compare the given test statistics to the critical value: The test statistic that is greater than the critical value and furthest from it will have the smallest probability of falling into the rejection region by chance, thus minimizing the Type I error
step 5
Select the correct answer: Among the given options, 4.56 is the test statistic that is greater than the critical value of 2.45 and is furthest from it
Answer
D
Key Concept
Minimizing Type I error in hypothesis testing
Explanation
To minimize the chance of making a Type I error, the test statistic should be greater than the critical value and as far away from it as possible
Multiple Choice: The one-tailed test keeps the rejection region ____ of the sampling distribution of a test statistics. A: in two tails B: in one tail C: in the lower tail D: in the upper tail
Solution by Steps
step 1
Understand the concept of a one-tailed test: A one-tailed test is a statistical hypothesis test in which the rejection region for the test statistic is located entirely in one tail of the sampling distribution
step 2
Determine the direction of the test: Depending on the alternative hypothesis, the rejection region could be in the lower tail (for testing if a parameter is less than a certain value) or in the upper tail (for testing if a parameter is greater than a certain value)
step 3
Identify the correct answer: Since the question specifies a one-tailed test but does not specify the direction of the test, the correct answer must indicate that the rejection region is in one tail, without specifying which tail
Answer
B
Key Concept
One-tailed tests have the rejection region in one tail of the distribution.
Explanation
In a one-tailed test, the rejection region is placed either in the lower tail or the upper tail of the sampling distribution, depending on the direction of the alternative hypothesis, but not in both tails.
Multiple CHOICE: A null hypothesis is not rejected if the computed value of _____ falls within the non-rejection region; otherwise, it is rejected. Group of answer choices A: population parameter B: sample C: test statistics D: sample statistic
Solution by Steps
step 1
Understand the concept of hypothesis testing: In hypothesis testing, we compare a test statistic calculated from the sample data to a critical value or a non-rejection region to decide whether to reject the null hypothesis
step 2
Identify the correct term: The computed value that is compared to the critical value or non-rejection region is the test statistic, which is derived from the sample data
step 3
Choose the correct answer: The null hypothesis is not rejected if the computed value of the test statistic falls within the non-rejection region
Answer
C
Key Concept
Test statistic in hypothesis testing
Explanation
The null hypothesis is not rejected if the computed test statistic falls within the non-rejection region.
Multiple Choice: The t-distribution differs from the standard z-distribution in one of the following ways. Which one? Group of answer choices A: Continuous distribution B: Symmetrical C: Bell-shaped D: Family of distributions
Solution by Steps
step 1
Identify the characteristics that are common to both t-distribution and standard z-distribution: Continuous, symmetrical, and bell-shaped
step 2
Recognize the unique characteristic of the t-distribution that differentiates it from the standard z-distribution: It is a family of distributions that varies with the degrees of freedom
step 3
Match the unique characteristic to the answer choices provided
Answer
D
Key Concept
Family of distributions
Explanation
The t-distribution is a family of distributions that changes with the degrees of freedom, unlike the standard z-distribution which is one distribution.
Multiple Choice: In a hypothesis test, if the rejection region is increased, Group of answer choices A: it becomes easier to reject H0. B: the risk of a Type I error increases. C: the value of α increases. D: the significance level is higher. all of the other answers are correct.
Solution by Steps
step 1
Consider the definition of the rejection region in hypothesis testing. The rejection region is the range of values for which the null hypothesis H0 H_0 is rejected
step 2
Understand that the rejection region is determined by the significance level α \alpha , which is the probability of making a Type I error (rejecting a true null hypothesis)
step 3
Recognize that increasing the rejection region means that more values fall into the range where H0 H_0 is rejected, which directly increases the probability of a Type I error
step 4
Acknowledge that increasing the probability of a Type I error is equivalent to increasing the value of α \alpha and thus the significance level
Answer
B, C, D
Key Concept
Rejection region and significance level
Explanation
Increasing the rejection region in a hypothesis test increases the probability of a Type I error, which is the significance level α \alpha .
Multiple Choice: If the alternate hypothesis states that µ ≠ 4,000, where is the rejection region for the hypothesis test? Group of answer choices A: In the lower or left tail B: In the center C: In the upper or right tail D: In both tails
Solution by Steps
step 1
Identify the type of hypothesis test based on the alternate hypothesis Ha:μ≠4000 H_a: \mu \neq 4000
step 2
Recognize that μ≠4000 \mu \neq 4000 indicates a two-tailed test because the alternate hypothesis does not specify a direction of the difference
step 3
Determine the rejection regions for a two-tailed test, which are located in both the lower (left) and upper (right) tails of the distribution
Answer
***D***
Key Concept
Two-tailed hypothesis test
Explanation
When the alternate hypothesis specifies that a parameter is not equal to a certain value, the hypothesis test is two-tailed, with rejection regions in both tails of the distribution.
Multiple Choice: In a hypothesis test, if the rejection region is increased, Group of answer choices A: it becomes easier to reject H0. B: the risk of a Type I error increases. C: the value of α increases. D: the significance level is higher. E: all of the other answers are correct.
Solution by Steps
step 1
Understand the concept of the rejection region in hypothesis testing: The rejection region is the range of values for which the null hypothesis H0 H_0 is rejected
step 2
Recognize the relationship between the rejection region and Type I error: A Type I error occurs when H0 H_0 is true but is incorrectly rejected. Increasing the rejection region increases the probability of making a Type I error
step 3
Identify the role of alpha (α \alpha ) in hypothesis testing: The value of α \alpha is the probability of making a Type I error, and it is also known as the significance level of the test
step 4
Relate the increase in the rejection region to α \alpha : Increasing the rejection region means that α \alpha increases, which in turn raises the significance level
Answer
E
Key Concept
Rejection region and Type I error
Explanation
Increasing the rejection region in a hypothesis test increases the risk of a Type I error, the value of α \alpha , and the significance level.
Multiple Choice: Restaurateur Daniel Valentine is evaluating the feasibility of opening a restaurant in Richmond. The Chamber of Commerce estimates that 'Richmond families, on the average, dine out at least 3 evenings per week'. Daniel plans to test this hypothesis using a random sample of 81 Richmond families. His alternative hypothesis is __________. Group of answer choices A: µ ≤ 3 B: X bar < 3 C: µ ≥ 3 D: µ < 3
Solution by Steps
step 1
Identify the null hypothesis based on the Chamber of Commerce's estimate: H0:μ≥3 H_0: \mu \geq 3
step 2
Determine the alternative hypothesis that Daniel wants to test against the null hypothesis. Since he is testing the claim that families dine out "at least" 3 evenings, he is looking for evidence that they dine out less than that
step 3
Formulate the alternative hypothesis: H_1: \mu < 3
Answer
D
Key Concept
Alternative Hypothesis (H1)
Explanation
The alternative hypothesis is a statement that the parameter (in this case, the population mean µ) is different from the null hypothesis (H0) in a specific way. In this scenario, since the claim is that families dine out "at least" 3 evenings, the alternative hypothesis should reflect the possibility that they dine out less often, which is \mu < 3 .
multiple choice: A political scientist wants to validate that a candidate is currently carrying more than 60% of the vote in the state. Her assistants randomly sample 200 eligible voters in the state by telephone and only 90 declare that they support her candidate. The observed z-value for this problem is . Group of answer choices A: -4.33 B: 4.33 C: -4.24 D: -4.26
Solution by Steps
step 1
Define the null hypothesis (H0H_0) and alternative hypothesis (HaH_a): H0:p=0.60H_0: p = 0.60 and H_a: p > 0.60, where pp is the true proportion of voters who support the candidate
step 2
Calculate the sample proportion (p^\hat{p}): p^=number of voters supporting the candidatetotal number of voters sampled=90200=0.45\hat{p} = \frac{\text{number of voters supporting the candidate}}{\text{total number of voters sampled}} = \frac{90}{200} = 0.45
step 3
Calculate the standard error (SE) of the sampling distribution under the null hypothesis: SE=p(1−p)n=0.60×(1−0.60)200=0.60×0.40200=0.24200=0.03464SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.60 \times (1 - 0.60)}{200}} = \sqrt{\frac{0.60 \times 0.40}{200}} = \sqrt{\frac{0.24}{200}} = 0.03464
step 4
Calculate the observed z-value: z=p^−pSE=0.45−0.600.03464=−0.150.03464=−4.33z = \frac{\hat{p} - p}{SE} = \frac{0.45 - 0.60}{0.03464} = \frac{-0.15}{0.03464} = -4.33
Answer
A
Key Concept
Calculating the observed z-value in hypothesis testing
Explanation
The observed z-value is calculated by subtracting the hypothesized population proportion from the sample proportion, then dividing by the standard error of the sampling distribution.
Multiple Choice: It is claimed that in a bushel of peaches, fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the sample proportion? Group of answer choices A: 40 B: 0.40 C: 0.125 D: 0.10
Solution by Steps
step 1
To calculate the sample proportion, we divide the number of defective peaches by the total number of peaches in the sample. The formula for the sample proportion (p^ \hat{p} ) is p^=xn \hat{p} = \frac{x}{n} , where x x is the number of successes (defective peaches) and n n is the sample size
step 2
Using the given numbers: x=50 x = 50 defective peaches and n=400 n = 400 total peaches, we plug these values into the formula to get p^=50400 \hat{p} = \frac{50}{400}
step 3
Simplifying the fraction 50400 \frac{50}{400} gives us p^=0.125 \hat{p} = 0.125
Answer
C
Key Concept
Sample Proportion Calculation
Explanation
The sample proportion is calculated by dividing the number of successes in the sample by the total sample size.
Multiple Choice: Suppose you want to test H0: µ = 30 versus H1: µ < 30. Which of the following possible sample results based on a sample of size 36 gives the strongest evidence to reject H0 in favor of H1? Group of answer choices A: X bar = 26, S = 9 B: X bar = 28, S = 6 C: X bar = 27, S = 4 D: X bar = 32, S = 2
Solution by Steps
step 1
Calculate the z-value for each sample result using the formula: z=Xˉ−μ0Sn z = \frac{\bar{X} - \mu_0}{\frac{S}{\sqrt{n}}} where Xˉ \bar{X} is the sample mean, μ0 \mu_0 is the hypothesized population mean, S S is the sample standard deviation, and n n is the sample size
step 2
For option A, calculate the z-value: zA=26−30936=−41.5=−2.67 z_A = \frac{26 - 30}{\frac{9}{\sqrt{36}}} = \frac{-4}{1.5} = -2.67
step 3
For option B, calculate the z-value: zB=28−30636=−21=−2 z_B = \frac{28 - 30}{\frac{6}{\sqrt{36}}} = \frac{-2}{1} = -2
step 4
For option C, calculate the z-value: zC=27−30436=−323=−4.5 z_C = \frac{27 - 30}{\frac{4}{\sqrt{36}}} = \frac{-3}{\frac{2}{3}} = -4.5
step 5
For option D, calculate the z-value: Since the sample mean is greater than the hypothesized mean, it does not provide evidence against H0 H_0 in favor of H1 H_1 . Therefore, we do not need to calculate the z-value for option D
step 6
Identify the sample result with the smallest (most negative) z-value, as this provides the strongest evidence against H0 H_0 in favor of H1 H_1
Answer
The sample result with the smallest z-value is option C, with a z-value of -4.5.
C
Key Concept
Strongest evidence against the null hypothesis
Explanation
In a left-tailed test, the strongest evidence to reject the null hypothesis in favor of the alternative hypothesis is provided by the sample result with the most negative z-value.
Multiple Choice: A random sample of size 15 is selected from a normal population. The population standard deviation is unknown. Assume the null hypothesis indicates a two-tailed test and the researcher decided to use the 0.10 significance level. For what values of t will the null hypothesis not be rejected? Group of answer choices A: Between −1.761 and 1.761 B: Between −1.282 and 1.282 C: To the left of −1.645 or to the right of 1.645 D: To the left of −1.345 or to the right of 1.345
Solution by Steps
step 1
Identify the degrees of freedom for the t-distribution: df=n−1 df = n - 1 where n n is the sample size
step 2
Calculate the degrees of freedom for the given sample size: df=15−1=14 df = 15 - 1 = 14
step 3
Determine the critical t-values for a two-tailed test with a significance level of α=0.10 \alpha = 0.10
step 4
Look up the critical t-values in the t-distribution table for df=14 df = 14 and α/2=0.05 \alpha/2 = 0.05 for each tail
step 5
The critical t-values are approximately ±1.761 \pm1.761
Answer
The null hypothesis will not be rejected for t-values between −1.761 and 1.761.
A
Key Concept
Critical t-values for a two-tailed test
Explanation
The critical t-values determine the range for which the null hypothesis is not rejected in a two-tailed t-test at a given significance level. These values are found using the degrees of freedom and the desired significance level in a t-distribution table.
Multiple Choice: Using a 5% level of significance and a sample size of 25, what is the critical t-value for a null hypothesis, H0: µ = 100? Group of answer choices A: 2.064 B: 1.711 C: 1.708 D: 2.060
Solution by Steps
step 1
Identify the degrees of freedom for the t-distribution: df=n−1 df = n - 1 where n n is the sample size
step 2
Calculate the degrees of freedom using the given sample size: df=25−1=24 df = 25 - 1 = 24
step 3
Determine the critical t-value for a two-tailed test at a 5% level of significance using a t-distribution table or calculator
step 4
For a two-tailed test with df=24 df = 24 and a 5% level of significance, the critical t-value is approximately 2.064
Answer
The critical t-value is 2.064.
A
Key Concept
Critical t-value for a two-tailed test
Explanation
The critical t-value is found using the degrees of freedom and the level of significance from the t-distribution table or calculator.
Multiple Choice: The probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true is known as the ___________blank. Group of answer choices A: observed α B: test statistic C: margin of error D: critical value
Solution by Steps
step 1
Identify the correct statistical term that describes the probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true
step 2
Recognize that this probability is used to determine whether to reject the null hypothesis and is calculated after the test statistic has been determined
step 3
Recall that this probability is not the observed alpha level (which is the pre-determined significance level), not the test statistic (which is a value calculated from the sample data), and not the margin of error (which is related to confidence intervals)
step 4
Understand that the correct term is the probability of making a Type I error if the null hypothesis is true, which is known as the p-value
Answer
The correct term is the p-value.
A
Key Concept
p-value
Explanation
The p-value is the probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true. It is used in hypothesis testing to decide whether to reject the null hypothesis.
Multiple choice: Which of the following statements can be called a hypothesis? Group of answer choices A: There are nine justices on the Supreme Court of the United States. B: It is raining outside. C: Tomorrow is Friday. D: The mean time of all crosswalk signals in the downtown area is 22 seconds.
Solution by Steps
step 1
Identify the characteristics of a hypothesis in statistics. A hypothesis is a statement that can be tested statistically, often about a population parameter
step 2
Evaluate each answer choice to determine if it represents a testable statement about a population parameter
step 3
Choice A is a fact and not a testable hypothesis
step 4
Choice B is an observation and not a testable hypothesis
step 5
Choice C is a statement about time and not a testable hypothesis
step 6
Choice D is a statement about a population parameter (mean time of crosswalk signals) that can be tested using statistical methods
Answer
D
Key Concept
Hypothesis in Statistics
Explanation
A hypothesis is a testable statement about a population parameter that can be evaluated using statistical methods.
Multiple Choice: You have created a 95% confidence interval for with the result 10 ≤ µ ≤ 15. What decision will you make if you test H0: µ = 16 versus H1: µ ≠ 16 at α = 0.05? Group of answer choices A: Reject H0 in favor of H1. B: Fail to reject H0 in favor of H1. C: Reject H1 in favor of H0. D: Do not reject H0 in favor of H1.
Solution by Steps
step 1
Interpret the confidence interval: The 95% confidence interval for the population mean μ \mu is given as 10≤μ≤15 10 \leq \mu \leq 15
step 2
Compare the hypothesized mean to the confidence interval: The null hypothesis H0:μ=16 H_0: \mu = 16 proposes a population mean that is not contained within the 95% confidence interval
step 3
Make a decision based on the confidence interval: Since the hypothesized mean μ=16 \mu = 16 is outside of the 95% confidence interval, we have sufficient evidence to reject H0 H_0 at the α=0.05 \alpha = 0.05 significance level
Answer
A
Key Concept
Confidence Interval and Hypothesis Testing Relationship
Explanation
If the hypothesized mean is not within the confidence interval, we reject the null hypothesis.
Multiple Choice: A hypothesis regarding the weight of newborn infants at a community hospital is that the mean is 6.5 pounds. A sample of seven infants is randomly selected and their weights at birth are recorded as 9.2, 5.8, 7.4, 7.0, 5.3, 9.0, and 8.9 pounds. If α = 0.200, what is the critical value? The population standard deviation is unknown. Group of answer choices A: ± 1.358 B: ± 0.936 C: 0 D: ± 1.440
Solution by Steps
step 1
Determine the appropriate distribution to use: Since the sample size is small (n=7) and the population standard deviation is unknown, we use the t-distribution
step 2
Find the degrees of freedom for the t-distribution: The degrees of freedom (df) is equal to the sample size minus one. df=n−1=7−1=6 df = n - 1 = 7 - 1 = 6
step 3
Identify the significance level: The given significance level is α=0.200 \alpha = 0.200 . For a two-tailed test, we divide the significance level by 2 to find the critical value for each tail. α/2=0.200/2=0.100 \alpha/2 = 0.200/2 = 0.100
step 4
Look up the critical t-value: Using a t-distribution table or calculator, find the t-value that corresponds to α/2=0.100 \alpha/2 = 0.100 with 6 degrees of freedom
step 5
Provide the critical t-value: The critical t-value for df=6 df = 6 and α/2=0.100 \alpha/2 = 0.100 is approximately ±1.440
Answer
D
Key Concept
Critical t-value
Explanation
The critical t-value is determined using the t-distribution with the appropriate degrees of freedom and significance level for a two-tailed test.
© 2023 AskSia.AI all rights reserved