91Ó°ÊÓ

The sample proportion is a valuable concept in statistics, especially in hypothesis testing. Essentially, the sample proportion is an estimate of the actual proportion of a population. In this exercise, out of 400 students, 289 reported not having enough sleep. The sample proportion, denoted by \( \hat{p} \), is calculated as the number of successes (in this case, students not getting enough sleep) divided by the total sample size.

• Formula: \( \hat{p} = \frac{x}{n} \)
• Where \( x \) is the number of successes, and \( n \) is the total number of observations.

Applying this, we have \( \hat{p} = \frac{289}{400} = 0.7225 \). This result means that approximately 72.25% of the sampled students reported insufficient sleep.

Understanding the sample proportion is crucial because it serves as the foundation for calculating other important statistics like standard error and test statistics in hypothesis testing.

The standard error (SE) is a measure of the variability or dispersion of a sample statistic, like the sample proportion, from the true population proportion. It gives us an idea of how much sampling variability there is likely to be.

• It is an essential component in hypothesis testing as it helps in constructing confidence intervals and evaluating the significance of tests.

For sample proportions, the standard error is calculated using the formula:
\[ SE = \sqrt{\frac{p(1-p)}{n}} \]
Where:

• \( p \) is the assumed population proportion under the null hypothesis.
• \( n \) is the number of observations in the sample.

In this exercise, \( p = 0.5 \) (50%), and \( n = 400 \), hence:
\[ SE = \sqrt{\frac{0.5 \times (1 - 0.5)}{400}} = 0.025 \]
This result indicates the standard deviation of the sample proportion, assuming that the true proportion of the population is 0.5.

The Z-Test is a type of statistical test that is used to determine whether there is a significant difference between sample and population means or proportions. It is particularly useful when the sample size is large.

• In hypothesis testing, a Z-test can compare the sample proportion against a known population proportion to see if the difference is significant.

The formula for the Z-statistic in the context of testing a sample proportion against a population proportion is:
\[ z = \frac{\hat{p} - p}{SE} \]
Where:

• \( \hat{p} \) is the sample proportion.
• \( p \) is the hypothesized population proportion.
• \( SE \) is the standard error of the sample proportion.

For the exercise, with \( \hat{p} = 0.7225 \), \( p = 0.5 \), and a standard error of 0.025, the Z-statistic is calculated as:
\[ z = \frac{0.7225 - 0.5}{0.025} = 8.9 \]
This high value suggests that the sample proportion is significantly different from the hypothesized population proportion.

The null hypothesis is a fundamental concept in hypothesis testing. It is a statement that there is no effect or no difference, and it serves as the starting assumption in statistical testing.

• The null hypothesis is symbolized by \( H_0 \), and is often the hypothesis that a researcher or scientist aims to test or disprove.

In this exercise, the null hypothesis is that the proportion of students who do not get enough sleep is equal to 50%, i.e., \( p = 0.5 \).

When conducting a hypothesis test, we determine whether there is sufficient statistical evidence in the sample to reject the null hypothesis.
Decision-making involves comparing the calculated test statistic (like the Z-score) with critical values.
If the test statistic falls in the rejection region beyond the critical values, the null hypothesis is rejected in favor of the alternative hypothesis, indicating a statistically significant difference.

In this case, since the calculated Z-value (8.9) exceeded the critical value (around 2.576 for a 0.01 significance level), the null hypothesis was rejected. This means the sample provides strong evidence that the proportion of students not getting enough sleep is different from 50%.