Statistics: Test of Significance for Proportion

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Questions and Answers

What conclusion can be drawn if the p-value is greater than the significance level alpha?

The null hypothesis is rejected.

The alternative hypothesis is accepted.

There is sufficient evidence for the alternative hypothesis.

The null hypothesis is accepted. (correct)

In Mendel's pea experiment, what was the actual ratio of smooth to wrinkled peas in the second generation?

75:25

3:1

5474:1850 (correct)

75:15

What was the significance level used in the pea experiment?

0.01

0.15

0.05 (correct)

0.10

What does a larger sample size typically do to the likelihood of rejecting the null hypothesis?

Makes it easier to reject. Signup and view all the answers

What was the decision regarding the null hypothesis in the pea experiment?

Fail to reject the null hypothesis. Signup and view all the answers

What was the p-value obtained from Mendel's data analysis?

0.610 Signup and view all the answers

What does it mean if the evidence supports the null hypothesis?

There is no significant difference from the expected ratio. Signup and view all the answers

What would be an effect of decreasing the sample size in a hypothesis test?

Decrease the likelihood of rejecting the null hypothesis. Signup and view all the answers

Which statement accurately describes the results of the Mendel's pea crossing experiment at a significance level of 0.05?

The data supports the hypothesis of 75% smooth trait occurrence. Signup and view all the answers

In hypothesis testing, what does alpha represent?

The level of significance set for the experiment. Signup and view all the answers

What is the null hypothesis (H0) when testing if a majority of constituents favour a proposal?

H0 : p = 0.5 Signup and view all the answers

Which condition must be checked for the test of significance to be valid?

Both np0 and n(1-p0) must be at least 10. Signup and view all the answers

In the context of a significance test, what does a P-value signify?

The probability of observing data as extreme as the test statistic under the null hypothesis. Signup and view all the answers

What alternative hypothesis (Ha) is being tested if a pharmaceutical company claims that less than 20% of patients experience side effects?

Ha: p < 0.2 Signup and view all the answers

How do you make a decision regarding the null hypothesis in a hypothesis test?

You assess whether the p-value is less than or equal to the significance level α. Signup and view all the answers

What is the correct formulation of a two-sided alternative hypothesis?

Ha: p ≠ p0 Signup and view all the answers

What is implied if you reject the null hypothesis in a statistical test?

You have found evidence to support the alternative hypothesis. Signup and view all the answers

If both np0 and n(1-p0) are less than 10, what does it indicate?

The sample size is too small for significance testing. Signup and view all the answers

In hypothesis testing, what does the null value (p0) represent?

The hypothesized proportion of the population. Signup and view all the answers

What could be the consequence of using a biased sample in hypothesis testing?

The test may yield misleading results about the population. Signup and view all the answers

Study Notes

Test of Significance for a Proportion

Steps in a significance test:
- State the null and alternative hypotheses.
- Check conditions and calculate the test statistic.
- Find the P-value using the appropriate distribution.
- Make a decision and state the conclusion in the context.

Hypotheses

Null hypothesis (H0): The proportion of interest (p) is equal to a specific value (p0).
Alternative hypothesis (Ha): The proportion of interest (p) is different from the null value (p0). This can be one-sided (right-sided or left-sided) or two-sided.

Examples

Parliamentarian voting: H0: p = 0.5 (50% of constituents favor the proposal), Ha: p > 0.5 (more than 50% favor the proposal).
New drug side effects: H0: p = 0.2 (20% of patients experience side effects), Ha: p < 0.2 (less than 20% experience side effects).
Children raised by grandparents: H0: p = 0.05 (5% of children are raised by grandparents), Ha: p ≠ 0.05 (the proportion has changed).

Condition Check

Random sample: The sample should be a representative selection from the population.
Sample size: The sample size must be large enough to ensure the sampling distribution is approximately normal.
- Check if both np0 and n(1-p0) are at least 10 where n is the sample size and p0 is the null value.

Test Statistic

The test statistic measures how far the observed sample proportion is from the null value.

P-value

The P-value is the probability of observing a sample proportion as extreme as the one obtained, assuming the null hypothesis is true.

Decision and Conclusion

Reject null hypothesis: If the P-value is less than or equal to the significance level (α), reject the null hypothesis. There is sufficient evidence to support the alternative hypothesis.
Fail to reject null hypothesis: If the P-value is greater than α, fail to reject the null hypothesis. There is not enough evidence to support the alternative hypothesis.

Effect of Sample Size

Larger sample: Increasing the sample size makes it easier to reject the null hypothesis for a given effect size.

Mendel's Peas Example

Test: Whether the data from smooth and wrinkled peas in the second generation (F2) agrees with the expected proportion of 75% for the dominant trait (smooth).
Hypotheses: H0: p = 0.75, Ha: p ≠ 0.75
Observed sample proportion: 5474 smooth / (5474 + 1850) = 0.747
P-value: 0.610 (calculated using the appropriate statistical test)
Conclusion: Since the P-value (0.610) is greater than the significance level (0.05), we fail to reject the null hypothesis. The data supports the conclusion that 75% of the F2 generation exhibits the dominant trait.

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Description

This quiz focuses on the steps involved in conducting a significance test for proportions, including formulating hypotheses and calculating test statistics. You will explore examples such as voting behavior and drug side effects to apply statistical concepts effectively.