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Questions and Answers
In constructing a confidence interval for a population proportion, which step involves confirming that there are at least 10 observed successes and failures?
In constructing a confidence interval for a population proportion, which step involves confirming that there are at least 10 observed successes and failures?
- Ensuring the sampling distribution is approximately normal. (correct)
- Identifying the critical value for the desired level of confidence.
- Constructing the interval using the formula $\hat{p} \pm z^* \times SE$.
- Interpreting the confidence interval in the context of the problem.
Which calculation is needed to determine if the sampling distribution is normally distributed when successes and failures are not explicitly given?
Which calculation is needed to determine if the sampling distribution is normally distributed when successes and failures are not explicitly given?
- Computing $n\hat{p}$ and $n(1 - \hat{p})$ to check for normality. (correct)
- Interpreting the confidence interval in the problem's context.
- Finding the square root of $\hat{p}(1 - \hat{p})/n$ to determine standard error.
- Calculating the critical value $z^*$ for the confidence level.
A researcher calculates a 95% confidence interval for the proportion of students who own a pet and finds the interval to be (0.20, 0.30). Which of the following is the correct interpretation?
A researcher calculates a 95% confidence interval for the proportion of students who own a pet and finds the interval to be (0.20, 0.30). Which of the following is the correct interpretation?
- The true proportion of students who own a pet is guaranteed to be between 0.20 and 0.30.
- We are 95% confident that the interval (0.20, 0.30) captures the true proportion of students who own a pet. (correct)
- 95% of the students own a pet, and their proportion lies between 0.20 and 0.30.
- There is a 95% probability that the true proportion falls within the interval (0.20, 0.30).
A study aims to estimate the proportion of defective products manufactured in a factory. A random sample of 200 products is inspected, and 10 are found to be defective. What is the sample proportion $\hat{p}$?
A study aims to estimate the proportion of defective products manufactured in a factory. A random sample of 200 products is inspected, and 10 are found to be defective. What is the sample proportion $\hat{p}$?
A polling agency wants to determine the proportion of voters who support a particular candidate. They take a random sample of 400 voters. How does increasing the sample size to 1600 voters affect the width of the confidence interval, assuming the sample proportion remains the same?
A polling agency wants to determine the proportion of voters who support a particular candidate. They take a random sample of 400 voters. How does increasing the sample size to 1600 voters affect the width of the confidence interval, assuming the sample proportion remains the same?
What is the purpose of hypothesis testing for a population proportion?
What is the purpose of hypothesis testing for a population proportion?
In hypothesis testing for a single proportion, what does the null hypothesis ($H_0$) typically state?
In hypothesis testing for a single proportion, what does the null hypothesis ($H_0$) typically state?
What is the purpose of calculating a test statistic in hypothesis testing?
What is the purpose of calculating a test statistic in hypothesis testing?
In hypothesis testing, the $p$-value is the probability of:
In hypothesis testing, the $p$-value is the probability of:
A researcher conducts a hypothesis test with a significance level of $\alpha = 0.05$. If the $p$-value is 0.03, what is the correct decision?
A researcher conducts a hypothesis test with a significance level of $\alpha = 0.05$. If the $p$-value is 0.03, what is the correct decision?
What condition(s) must be met to proceed with a z-test for a proportion?
What condition(s) must be met to proceed with a z-test for a proportion?
A researcher is testing the hypothesis that the proportion of adults who prefer coffee over tea is greater than 0.5. They collect data and calculate a test statistic with a corresponding p-value of 0.08. Which of the following statements is correct if they are using significance level $\alpha = 0.05$?
A researcher is testing the hypothesis that the proportion of adults who prefer coffee over tea is greater than 0.5. They collect data and calculate a test statistic with a corresponding p-value of 0.08. Which of the following statements is correct if they are using significance level $\alpha = 0.05$?
A geneticist claims that 75% of offspring will have red flowers. In a test with 100 seeds, only 63 resulted in red flowers. If the test statistic $z = -2.77$ is calculated, and the alternative hypothesis is $H_a : p \neq 0.75$, what does the $p$-value represent?
A geneticist claims that 75% of offspring will have red flowers. In a test with 100 seeds, only 63 resulted in red flowers. If the test statistic $z = -2.77$ is calculated, and the alternative hypothesis is $H_a : p \neq 0.75$, what does the $p$-value represent?
Under what condition can results from hypothesis testing and confidence intervals be directly compared?
Under what condition can results from hypothesis testing and confidence intervals be directly compared?
If a claimed value falls outside the confidence interval, which of the following is true regarding a two-sided hypothesis test with significance level $\alpha$?
If a claimed value falls outside the confidence interval, which of the following is true regarding a two-sided hypothesis test with significance level $\alpha$?
When comparing two proportions, what parameter are we defining?
When comparing two proportions, what parameter are we defining?
In hypothesis testing for two proportions ($p_1$ and $p_2$), the null hypothesis is often $H_0: p_1 - p_2 = 0$. What does this null hypothesis imply?
In hypothesis testing for two proportions ($p_1$ and $p_2$), the null hypothesis is often $H_0: p_1 - p_2 = 0$. What does this null hypothesis imply?
When conducting a hypothesis test comparing two proportions, what conditions must be met regarding the sample sizes and the number of successes and failures?
When conducting a hypothesis test comparing two proportions, what conditions must be met regarding the sample sizes and the number of successes and failures?
In a two-sample proportion test, what is the purpose of using a pooled proportion?
In a two-sample proportion test, what is the purpose of using a pooled proportion?
When constructing a confidence interval for the difference between two proportions, what does it indicate if the interval contains zero?
When constructing a confidence interval for the difference between two proportions, what does it indicate if the interval contains zero?
You are comparing the proportions of customers who prefer online shopping versus in-store shopping between two different age groups. The 95% confidence interval for the difference in proportions is (-0.05, 0.15). How should you interpret this interval?
You are comparing the proportions of customers who prefer online shopping versus in-store shopping between two different age groups. The 95% confidence interval for the difference in proportions is (-0.05, 0.15). How should you interpret this interval?
What type of data is typically used in a goodness-of-fit test?
What type of data is typically used in a goodness-of-fit test?
In a goodness-of-fit test, the null hypothesis ($H_0$) typically states that:
In a goodness-of-fit test, the null hypothesis ($H_0$) typically states that:
A researcher wants to determine if the distribution of eye colors in a sample matches the known distribution in the general population. They perform a goodness-of-fit test. What does a large test statistic value suggest?
A researcher wants to determine if the distribution of eye colors in a sample matches the known distribution in the general population. They perform a goodness-of-fit test. What does a large test statistic value suggest?
What is the formula to find the expected count for each category?
What is the formula to find the expected count for each category?
A goodness-of-fit test yields a test statistic of 15.0 with 4 degrees of freedom. How is the p-value determined?
A goodness-of-fit test yields a test statistic of 15.0 with 4 degrees of freedom. How is the p-value determined?
In a chi-square goodness-of-fit test, how are degrees of freedom (df) calculated if there are k categories?
In a chi-square goodness-of-fit test, how are degrees of freedom (df) calculated if there are k categories?
A company claims that their candies are distributed in the following proportions: 30% red, 20% blue, 25% green, and 25% yellow. A sample of 300 candies yields the following counts: 80 red, 70 blue, 80 green, and 70 yellow. Which of the following is the correct conclusion if the goodness-of-fit test has a test statistic with a corresponding $p$-value is 0.0005?
A company claims that their candies are distributed in the following proportions: 30% red, 20% blue, 25% green, and 25% yellow. A sample of 300 candies yields the following counts: 80 red, 70 blue, 80 green, and 70 yellow. Which of the following is the correct conclusion if the goodness-of-fit test has a test statistic with a corresponding $p$-value is 0.0005?
What is a two-way table used for?
What is a two-way table used for?
What is the null hypothesis for a test of independence in a two-way table?
What is the null hypothesis for a test of independence in a two-way table?
What does the expected count in a two-way table represent, assuming the null hypothesis is true?
What does the expected count in a two-way table represent, assuming the null hypothesis is true?
If $R$ is the number of rows and $C$ is the number of columns in a two-way table, how is the degrees of freedom (df) calculated for a chi-square test of independence?
If $R$ is the number of rows and $C$ is the number of columns in a two-way table, how is the degrees of freedom (df) calculated for a chi-square test of independence?
What value of $x^2$ would lead to the rejection of the null hypothesis?
What value of $x^2$ would lead to the rejection of the null hypothesis?
A researcher analyzes a two-way table and calculates a chi-square test statistic of 8.5 with 2 degrees of freedom. If their significance level is $\alpha = 0.05$, estimate the p-value based on knowing the higher the $x^2$, the lower p-value. What conclusion can they draw?
A researcher analyzes a two-way table and calculates a chi-square test statistic of 8.5 with 2 degrees of freedom. If their significance level is $\alpha = 0.05$, estimate the p-value based on knowing the higher the $x^2$, the lower p-value. What conclusion can they draw?
Which statement is correct about the expected count of the cell?
Which statement is correct about the expected count of the cell?
A survey asks respondents whether they support a new policy and categorizes them by gender (male, female). The data is displayed in a two-way table. What are the appropriate hypotheses for analyzing this data?
A survey asks respondents whether they support a new policy and categorizes them by gender (male, female). The data is displayed in a two-way table. What are the appropriate hypotheses for analyzing this data?
Flashcards
Confidence Interval
Confidence Interval
A range of plausible values for a population parameter.
Step 1: Confidence Interval
Step 1: Confidence Interval
Identify given information and determine the critical value.
Step 2: Normality Check
Step 2: Normality Check
Check for normal distribution by confirming at least 10 successes and failures.
Step 3: Construct interval
Step 3: Construct interval
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Standard Error (SE)
Standard Error (SE)
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Step 4: Interpretation
Step 4: Interpretation
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Hypotheses
Hypotheses
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Null Hypothesis (H0)
Null Hypothesis (H0)
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Alternative Hypothesis (Ha)
Alternative Hypothesis (Ha)
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Test Statistic
Test Statistic
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Conditions for Hypothesis test
Conditions for Hypothesis test
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P-value
P-value
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Draw conclusions
Draw conclusions
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P-value < α
P-value < α
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Confidence Intervals & Hypothesis Test
Confidence Intervals & Hypothesis Test
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Statistically significant result
Statistically significant result
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Comparing Two Proportions
Comparing Two Proportions
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Pooled Proportion
Pooled Proportion
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Goodness-of-Fit
Goodness-of-Fit
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Expected Count
Expected Count
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Degrees of freedom for Goodness of fit
Degrees of freedom for Goodness of fit
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Two-way table
Two-way table
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Study Notes
Inference for a Single Proportion
- Construct confidence intervals to estimate a plausible range of values for a population parameter.
Steps to Constructing Confidence Intervals
- Step 1: Identify the sample proportion (p̂), sample size (n), and the critical value (z*) for the desired confidence level.
- Step 2: Verify the sampling distribution is normally distributed by ensuring at least 10 observed successes and failures.
- Step 3: Construct the interval using the formula: p̂ ± z* multiplied by SE .
- Step 4: Interpret the confidence interval in the context of the problem.
Hypothesis Testing procedure for a Single Proportion
- Step 1: State the hypotheses (Null = Ho : p = po, Alternative = Ha : p < po, Ha : p > po, Ha : p ̸= po).
- Step 2: Calculate the test statistic to measure the deviation between the data and the claimed value.
- Step 3: Calculate the p-value, depending on the form of the alternative hypothesis.
- Step 4: Make conclusions in the context of the problem.
P Value
- Ha: p < po (left-sided test) use the table value
- Ha: p > po (right-sided test) use 1 - table value
- Ha: p ̸= po (two-sided test) Look up the negative z value, then multiply the table value by 2
Decisions On Hypothesis Testing
- If the p-value is less than or equal to α, there is enough evidence to reject the null hypothesis
- If the p-value is greater than α, the null hypothesis is not rejected.
- Conclusion indicate whether to reject or fail to reject Ho.
Additional Notes on Proportions
- Using a two-sided hypothesis test with significance level α yields the same conclusions as a 100(1 − α)% confidence interval.
- Rejecting the null hypothesis for a two-sided test is equivalent to the claimed value not being in the confidence interval.
- Failing to reject the null hypothesis for a two-sided test is equivalent to the claimed value being within the confidence interval.
Two Proportions
- Analyze the difference (p1 − p2) between two independent population proportions using hypothesis tests and confidence intervals.
Steps to Comparing Two Proportions
- Step 1: State the hypotheses - Ho : p1 − p2 = 0 with alternative hypotheses (Ha : p1 − p2 < 0, Ha : p1 − p2 > 0, Ha : p1 − p2 ̸= 0).
- Step 2: Find an estimator for p1 − p2 and its standard error.
- Step 3: Calculate the p-value
- Step 4: Make conclusions in the context of the problem.
Conditions for comparing proportions
- During hypothesis testing, ensure the proportions are the same, using a pooled proportion to adjust the standard error.
- When calculating the test statistic, the formula required is Z score
- Both sample sizes should have at least ten successes and failures
- Note that the order doesn't matter, as long as it remains consistent throughout the problem
Goodness-of-Fit Test
- Explores questions about proportions from several groups using the χ2 distribution, often with one-way tables.
Steps for Goodness-of-Fit Test
- Step 1: State the hypotheses
- Step 2: Determine a meaningful expected count for each category.
- Step 3: Calculate the Test Statistic: The test statistic is a measure of how well the observed data fits the expected distribution
- Step 4: Look at observed vs expected counts
Chi Square Test
- X² is defined as the sum of squared independent standard normal random variables and determined by the parameter ν
- P − value = P (χ2 > χ2o )
- The degrees of freedom are based on k − 1
Rules for Comparing Alpha Values
- Alpha = standard value
- If p − value ≤ α, there is enough evidence to reject the null hypothesis
- If p − value > α, there is not enough evidence to reject the null hypothesis.
Two-Way Tables
- Cell counts are computed as : Row i Total multiplied by Column j Total divided by Table Total
Statistic for a Chi Square test
- The test statistic for a chi square test of independence mirrors the goodness of fit test, computing the sum of squared differences
- Conditions for the test, with more than 5 expected counts in each cell.
- Degrees of freedom, computed as is: d.f. = (R-1) multiplied by (C-1)
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