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Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. In a random sample of males, it was found that 23 write with their left hands and 217 do not. In a random sample of females, it was found that 65 write with their left hands and 455 do not (based on data from "The Left-Handed: Their Sinister History," by Elaine Fowler costas, Education 91影视 Information Center, Paper 399519 ). We want to use a 0.01 significance level to test the claim that the rate of left-handedness among males is less than that among females. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Based on the results, is the rate of left-handedness among males less than the rate of left handedness among females?

Step by step solution

01

Define Hypotheses

Define the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis states that the proportion of left-handed males is equal to the proportion of left-handed females. The alternative hypothesis claims that the proportion of left-handed males is less than the proportion of left-handed females.H0: p_m = p_fH1: p_m < p_f

02

Calculate Sample Proportions

Calculate the sample proportions for both males and females.p_m = 23 / 240 = 0.0958p_f = 65 / 520 = 0.125

03

Calculate Test Statistic

Use the pooled sample proportion to calculate the test statistic.\[ \text{Pooled proportion} = \frac{23+65}{240+520} = 0.1167\]The test statistic (z) is given by:\[ z = \frac{p_m - p_f}{\text{Standard Error}} = \frac{0.0958 - 0.125}{\text{SE}}\]Where the standard error (SE) is:\[ SE = \text{sqrt}(0.1167 \times (1 - 0.1167) \times (\frac{1}{240} + \frac{1}{520})) = 0.0213\]Then the test statistic is:\[ z = \frac{0.0958 - 0.125}{0.0213} = -1.374\]

04

Find Critical Value or P-value

Find the critical value or P-value for the given significance level. For a one-tailed test with a significance level of 0.01, the critical value is -2.33. Alternatively, look up the P-value for the test statistic z = -1.374 in standard normal distribution tables. The P-value is approximately 0.085.

05

State Conclusion

Compare the test statistic to the critical value or the P-value to the significance level (0.01).Since -1.374 > -2.33 or P-value = 0.085 > 0.01, we fail to reject the null hypothesis.

06

Confidence Interval Approach

Calculate the confidence interval for the difference in proportions. For a 99% confidence interval, the margin of error (ME) is:\[ ME = Z_{\text{critical}} \times SE = 2.33 \times 0.0213 = 0.0496\]Thus, the confidence interval for the difference ( p_m - p_f ) is:\[ 0.0958 - 0.125 \text{ 卤 } 0.0496 = [-0.0788, -0.020]\]Since the interval includes negative values, it suggests the possibility of males being less left-handed, but it includes 0 which means not definitive.

07

Conclusion about Original Claim

Based on both the hypothesis test and confidence interval, there is not enough evidence to conclude that the rate of left-handedness among males is less than that among females at the 0.01 significance level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

null hypothesis

A hypothesis test always starts with the **null hypothesis (H鈧�)**. This is a statement that there is no effect or no difference, and it is what we test against. In this exercise, our null hypothesis states that the proportion of left-handed males is equal to the proportion of left-handed females. Symbolically, this is written as:
**H鈧�: p鈧� = p_f**
We assume the null hypothesis to be true until the data provides strong evidence against it.

alternative hypothesis

The **alternative hypothesis (H鈧�)** is what you might believe to be true or hope to prove true. This hypothesis is often what the researcher is interested in. In our case, we want to test if the proportion of left-handed males is less than that of left-handed females. Therefore, the alternative hypothesis is:
**H鈧�: p鈧� < p_f**
Understanding the alternative hypothesis helps to determine the direction of the test, whether it's one-tailed or two-tailed.

test statistic

The **test statistic** is a standardized value that helps to determine how far away our sample statistic is from the null hypothesis. For this problem, we use a z-test for proportions, which involves calculating a pooled sample proportion and then finding the standard error.
- Pooled proportion, \[\frac{23+65}{240+520} = 0.1167\]
- Standard error (SE), \[\text{sqrt}(0.1167 \times (1 - 0.1167) \times (\frac{1}{240} + \frac{1}{520})) = 0.0213\]
- Test statistic (z), \[\frac{0.0958 - 0.125}{0.0213} = -1.374\]
The test statistic tells us how many standard errors our sample proportion is from the null hypothesis proportion.

P-value

The **P-value** represents the probability of obtaining results as extreme as, or more extreme than, the ones observed, under the assumption that the null hypothesis is true. For our test statistic z = -1.374, we find the P-value:
The P-value here is approximately 0.085. This is compared against the significance level (伪), which in this exercise is 0.01. Since 0.085 > 0.01, we do not have enough evidence to reject the null hypothesis.
A smaller P-value (less than 伪) would indicate strong evidence against the null hypothesis, leading to its rejection.

confidence interval

A **confidence interval** gives a range of values within which we expect the true population parameter to lie. For a 99% confidence interval, calculate the margin of error (ME):
- Critical value for 99% CI, Z_{critical} = 2.33
- ME, \[\text{ME} = 2.33 \times 0.0213 = 0.0496\]
The confidence interval for the difference in proportions (p鈧� - p鈧�) is:
\[0.0958 - 0.125 \text{ 卤 } 0.0496 = [-0.0788, -0.020]\]
Because this interval includes negative values and zero, it suggests that while the rate of left-handedness among males is potentially less than among females, the evidence is not definitive. This supports our earlier conclusion that we do not have enough evidence to reject the null hypothesis.