A random sample of 140 males shows that 41 males have a certain minor blood disorder, while a random sample of 110 females shows that only 30 have the disorder. Can we conclude at a 0,01 level of significance that the proportion of men in the population with this blood disorder is significantly greater than the proportion of women with this disorder. Give the test statistic.
To determine if the proportion of men with the blood disorder is significantly greater than the proportion of women with the disorder, we can perform a hypothesis test using the two-proportion z-test.
Let's define the null and alternative hypotheses:
- Null Hypothesis: There is no significant difference in the proportion of men and women with the blood disorder.
- Alternative Hypothesis: The proportion of men with the blood disorder is significantly greater than the proportion of women.
Next, we need to calculate the test statistic using the formula:
z = (p1 - p2) / sqrt( p * (1 - p) * (1/n1 + 1/n2) )
Where:
- p1 is the proportion of men with the blood disorder (41/140)
- p2 is the proportion of women with the blood disorder (30/110)
- p is the pooled proportion (total number of successes / total number of trials)
- p = (x1 + x2) / (n1 + n2), where x1 = number of successes in the first sample, x2 = number of successes in the second sample, n1 = size of the first sample, n2 = size of the second sample
Calculating p, we have:
p = (41 + 30) / (140 + 110)
Now, we can substitute the values into the formula to calculate the test statistic:
z = ( (41/140) - (30/110) ) / sqrt( (p * (1 - p) * (1/140 + 1/110)) )
After calculating the test statistic, we can compare it to the critical value associated with a significance level of 0.01. If the test statistic falls in the rejection region, we reject the null hypothesis in favor of the alternative hypothesis.
Please note that I have used the given sample sizes and proportions in this explanation. To obtain the actual test statistic value, you need to substitute the provided sample numbers into the formula and calculate it.
To determine if the proportion of men with the blood disorder is significantly greater than the proportion of women with the disorder, we can perform a two-sample proportion z-test.
First, we need to calculate the sample proportions for both males and females:
For males:
Sample proportion (p1) = 41/140 = 0.293
For females:
Sample proportion (p2) = 30/110 = 0.273
Next, we determine the test statistic using the formula:
z = (p1 - p2) / √(p̂(1-p̂) * (1/n1 + 1/n2))
Where:
p̂ = (n1 * p1 + n2 * p2) / (n1 + n2)
n1 and n2 are the sample sizes for males and females, respectively.
For this case:
n1 = 140, n2 = 110
p̂ = (140 * 0.293 + 110 * 0.273) / (140 + 110) = 0.280
Now, we can calculate the test statistic:
z = (0.293 - 0.273) / √(0.280 * (1-0.280) * (1/140 + 1/110))
z ≈ 0.020 / √(0.280 * 0.720 * (0.0071 + 0.0091))
z ≈ 0.020 / √(0.280 * 0.720 * 0.0162)
z ≈ 0.020 / √(0.00873728)
z ≈ 0.020 / 0.0934525
z ≈ 0.214
The test statistic is approximately 0.214.
To determine if this test statistic is significant at a 0.01 level of significance, we compare it to the critical z-value for a 0.01 (α = 0.01) level of significance. We can use a standard normal distribution table or a calculator to find that the critical z-value for a two-tailed test at a 0.01 level of significance is approximately ±2.576.
Since 0.214 is within the range of -2.576 to +2.576, we fail to reject the null hypothesis. Thus, we cannot conclude at a 0.01 level of significance that the proportion of men with the blood disorder is significantly greater than the proportion of women with the disorder.
To determine whether the proportion of men in the population with the blood disorder is significantly greater than the proportion of women with the disorder, we can perform a hypothesis test.
Let's denote:
- p1: Proportion of men with the blood disorder in the population
- p2: Proportion of women with the blood disorder in the population
The null hypothesis (H0) assumes no difference between the proportions:
H0: p1 = p2
The alternative hypothesis (Ha) assumes a greater proportion of men with the blood disorder:
Ha: p1 > p2
Given the sample sizes and number of individuals with the blood disorder in each gender, we can calculate the sample proportions:
p̂1 = 41/140 ≈ 0.293
p̂2 = 30/110 ≈ 0.273
Now, we can perform a two-proportion z-test to determine the test statistic. The formula for the test statistic is:
z = (p̂1 - p̂2) / √((p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2))
where n1 and n2 are the respective sample sizes.
Substituting the values:
z = (0.293 - 0.273) / √((0.293 * (1 - 0.293) / 140) + (0.273 * (1 - 0.273) / 110))
Calculating this expression results in the test statistic z.
Once we have the test statistic, we can compare it to the critical value for a 0.01 level of significance. If the test statistic is greater than the critical value, we can reject the null hypothesis in favor of the alternative hypothesis.
Note: To provide the specific test statistic value and determine whether the null hypothesis can be rejected, the sample sizes would need to be given in the question.