Question

A university conducts a survey of 300 of its students, and 191 of the students say they do not get enough sleep. A hypothesis test is conducted to test whether the proportion of students who do not get enough sleep is greater than 50%. What is the value of the test statistic for this hypothesis test?

Answer 1

The test statistic for the hypothesis test to determine if the proportion of university students who do not get enough sleep is greater than 50% is approximately 4.73.

Step-by-step explanation:

To calculate the value of the test statistic for the hypothesis test where we are testing whether the proportion of students who do not get enough sleep is greater than 50%, we can use the formula for a test of a single population proportion:

z = (p' - p0) / sqrt(p0(1 - p0) / n)

Where:

• p' is the sample proportion
• p0 is the hypothesized population proportion
• n is the sample size

Here,

• p' = 191/300
• p0 = 0.50 (50%)
• n = 300

Substituting these values into the formula:

z = (191/300 - 0.50) / sqrt(0.50 × 0.50 / 300)

z = 0.1367 / sqrt(0.0008333)

z ≈ 0.1367 / 0.0289 ≈ 4.731

Therefore, the value of the test statistic for this hypothesis test is approximately 4.73.

Answer 2

The value of the test statistic for this hypothesis test is approximately 1.96.

Step-by-step explanation:

The test statistic is calculated using the formula:

z=(p−p^)p(1−p)nz=np(1−p)​

​(p−p^​)​

where pp is the assumed population proportion (in this case, 0.5 for a null hypothesis of no difference), p^p^​ is the sample proportion (191 out of 300 students), and nn is the sample size (300 students). The calculated test statistic is then compared to critical values from the standard normal distribution to determine the significance of the result.

In this context, the value of 1.96 corresponds to the critical value for a one-tailed test at a 5% significance level. If the calculated test statistic is greater than 1.96, it indicates that the proportion of students who do not get enough sleep is significantly greater than 50%, leading to the rejection of the null hypothesis.