To test the hypothesis that the mean time for students at this college to complete the entrance exam is greater than 82 minutes, we conduct a one-sample t-test since the sample size is small ($n = 8$) and the population standard deviation is not known.

Hypotheses

• Null Hypothesis ($H_0$): The mean time to complete the exam is 82 minutes or less. ($\mu \leq 82$)
• Alternative Hypothesis ($H_a$): The mean time to complete the exam is greater than 82 minutes. ($\mu > 82$)

This is a one-tailed t-test.

Given Data

• Sample mean ($\bar{x}$) = 94.88
• Sample standard deviation ($s$) = 21.44
• Sample size ($n$) = 8
• Significance level ($\alpha$) = 0.05
• Population mean under $H_0$ ($\mu_0$) = 82

Test Statistic

The formula for the $t$-statistic is:

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Substitute the given values:

$t = \frac{94.88 - 82}{21.44 / \sqrt{8}}$

Let me calculate this.import math

Given data

sample_mean = 94.88 hypothesized_mean = 82 sample_std_dev = 21.44 sample_size = 8

Calculate t-statistic

t_statistic = (sample_mean - hypothesized_mean) / (sample_std_dev / math.sqrt(sample_size)) t_statistic1.6991670413587181.699167041358718successThe calculated $t$-statistic is approximately $t = 1.70$.

Degrees of Freedom

The degrees of freedom ($df$) for this test is:

$df = n - 1 = 8 - 1 = 7$

Critical Value

For a one-tailed test at $\alpha = 0.05$ and $df = 7$, we find the critical value of $t$ using the $t$-distribution.

Let me calculate the critical value.import scipy.stats as stats

Significance level and degrees of freedom

alpha = 0.05 df = sample_size - 1

Critical t-value for one-tailed test

critical_t = stats.t.ppf(1 - alpha, df) critical_t1.8945786050613051.894578605061305successThe critical $t$-value for a one-tailed test at $\alpha = 0.05$ and $df = 7$ is approximately $t_{critical} = 1.895$.

Decision Rule

• If $t \geq t_{critical}$, reject the null hypothesis ($H_0$).
• If $t < t_{critical}$, fail to reject the null hypothesis.

Conclusion

The calculated $t$-statistic ($t = 1.70$) is less than the critical $t$-value ($t_{critical} = 1.895$).

Thus, we fail to reject the null hypothesis. At a significance level of 0.05, there is insufficient evidence to support the claim that the mean time for students at this college to complete the entrance exam is greater than 82 minutes.

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Do you want further explanation or have any questions?

Here are five related questions to deepen understanding:

1. What would happen if the sample size were larger, say 30?
2. How does changing the significance level affect the conclusion?
3. Can you explain why we use a $t$-test instead of a $z$-test?
4. What is the difference between one-tailed and two-tailed tests?
5. How would we test this hypothesis using a p-value approach?

Tip: Always compare the $t$-statistic to the critical value based on your chosen significance level to make accurate decisions.