Question

It is claimed that 90% of the students in STSA 1 class will pass their test. After the test a random sample of 50 students shows that 46 passed the test. Use an appropriate test on a 0,01 level of significance to investigate the claim. Determine the critical value.

Answer 1

To investigate the claim, we can use a hypothesis test. Let's set up the null and alternative hypotheses:

Null hypothesis (H0): The proportion of students passing the test is 90% (P = 0.9)
Alternative hypothesis (Ha): The proportion of students passing the test is not equal to 90% (P ≠ 0.9)

We will use the z-test for proportions since the sample size is large (n = 50) and the data is binomial (pass or fail). The test statistic for the z-test is calculated as:

z = (p̂ - P) / √(P * (1-P) / n)

Where p̂ is the sample proportion, P is the hypothesized proportion, and n is the sample size.

From the given data, p̂ (sample proportion) = 46/50 = 0.92.
P (hypothesized proportion) = 0.9.
n (sample size) = 50.

Calculating the test statistic:
z = (0.92 - 0.9) / √(0.9 * (1-0.9) / 50)
= 0.02 / √(0.09 / 50)
= 0.02 / √0.0018
= 0.02 / 0.0424264
= 0.472

To find the critical value, we need to determine the z-score corresponding to a 0.01 level of significance. For a two-tailed test, the critical value is z = ±2.575.

Therefore, the critical value for this test is 2.575.