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In a survey of 190 females who recently completed high​ school, ​70% were enrolled in college. In a survey of 160 males who recently completed high​ school, 60​% were enrolled in college. At ​a=0.09, can you reject the claim that there is no difference in the proportion of college enrollees between the two​ groups? Assume the random samples are independent. Complete parts​ (a) through​ (e).

User Sanne
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Answer:

Explanation:

To determine if we can reject the claim that there is no difference in the proportion of college enrollees between the two groups (females and males), we need to perform a hypothesis test.

Let:

p1 = Proportion of females enrolled in college

p2 = Proportion of males enrolled in college

We are given the following information:

n1 = Sample size for females = 190

n2 = Sample size for males = 160

x1 = Number of females enrolled in college = 0.70 * 190 = 133

x2 = Number of males enrolled in college = 0.60 * 160 = 96

(a) State the null and alternative hypotheses:

Null hypothesis (H0): p1 - p2 = 0 (There is no difference in the proportion of college enrollees between the two groups)

Alternative hypothesis (Ha): p1 - p2 ≠ 0 (There is a difference in the proportion of college enrollees between the two groups)

(b) Calculate the pooled sample proportion (p):

p = (x1 + x2) / (n1 + n2)

p = (133 + 96) / (190 + 160)

p = 229 / 350

p ≈ 0.6543

(c) Calculate the standard error (SE) of the difference in proportions:

SE = √[p * (1 - p) * (1/n1 + 1/n2)]

SE = √[0.6543 * (1 - 0.6543) * (1/190 + 1/160)]

SE ≈ √[0.6543 * 0.3457 * (0.005263 + 0.00625)]

SE ≈ √[0.227 * 0.011513]

SE ≈ √0.002614

SE ≈ 0.05113

(d) Calculate the test statistic (z-score):

z = (p1 - p2) / SE

z = (0.70 - 0.60) / 0.05113

z ≈ 1.954

(e) Conduct the hypothesis test at α = 0.09:

At α = 0.09 (or 9% significance level), we have a two-tailed test.

The critical z-value for a two-tailed test at α = 0.09 is approximately ±1.695.

Since the calculated z-score (1.954) falls outside the critical region of ±1.695, we can reject the null hypothesis.

Conclusion:

Based on the hypothesis test, we can reject the claim that there is no difference in the proportion of college enrollees between the two groups (females and males) at α = 0.09 significance level. There is evidence to suggest that there is a difference in the proportion of college enrollees between the two groups.

User Gtangil
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