it has been found that 85.6% of all enrolled college and university students in the U.S. are undergraduates. A random sample of 500 enrolled college students in a particular state revealed that 420 of them were undergraduates. Is there sufficient evidence to conclude that the population differs from the national percentages? Use a = 0.05
This is 1-Proportion Z-Test with p = 420 / 500 = 0.84
Ho: mean proportion = 0.856
Ha: mean proportion ≠ 0.856 [Since we are using ≠ , this is a two-tailed test]
z-statistic = (0.84 - 0.856) / sqrt[ (0.856)(1-0.856) / 500] = -1.019
p-value = (2)P(z < -1.019) = (2)(0.1541) = 0.3082
NOTE: we multiplied by 2 because this is a "two-tailed" test.
Conclusion: Since p-value > 0.05, do NOT reject the null hypothesis. There is significant evidence that the mean proportion of undergraduates is not different from 85.6%
hope that helped
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It has been found that 85.6% of all enrolled college and university students in tge united states are undergrates. A random sample of 500 enrolled college students in a paticular state revealed that 420 of them were undergrates. Is there sufficient evidence.to conclude that the proporation differs from the national percentage? Use x= 0.05
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Are you asking if there's enough evidence to conclude that the population of enrolled college students in this particular state are secretly all trained circus clowns? Because if that's the case, I'd say we definitely need more evidence than just the percentage of undergraduates in the population.
To determine whether there is sufficient evidence to conclude that the population differs from the national percentages, we can perform a hypothesis test.
Step 1: State the hypotheses.
- Null hypothesis (H0): The population proportion is equal to the national percentage of 85.6%.
- Alternative hypothesis (Ha): The population proportion is different from the national percentage of 85.6%.
Step 2: Determine the significance level (α).
The significance level (α) is given as 0.05 (5%).
Step 3: Compute the test statistic.
We can use the Z-test statistic for proportions to compare the sample proportion to the hypothesized population proportion. The formula for the Z-test statistic is:
Z = (p̂ - p) / sqrt(p * (1 - p) / n)
Where:
- p̂ represents the sample proportion (420/500 = 0.84)
- p represents the hypothesized population proportion (0.856)
- n represents the sample size (500)
Let's plug in the values:
Z = (0.84 - 0.856) / sqrt(0.856 * (1 - 0.856) / 500)
Step 4: Determine the critical value(s).
Since the alternative hypothesis is "different," we will perform a two-tailed test. The critical value(s) can be found using a Z-table or a statistical software. For a significance level of 0.05 (two-tailed test), the critical value is approximately ±1.96.
Step 5: Make a decision.
If the calculated test statistic falls within the critical region (outside the critical values), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Let's calculate the value of Z and compare it with the critical values:
Z = (0.84 - 0.856) / sqrt(0.856 * (1 - 0.856) / 500)
= -2.36
Since -2.36 is outside the ±1.96 critical region, we can reject the null hypothesis.
Step 6: State the conclusion.
Based on the statistical analysis, there is sufficient evidence to conclude that the population proportion of college students in the particular state differs from the national percentage of 85.6% at a significance level of 0.05.