A survey found that 36% of students still enjoy doing research in a library. A random sample of 326 students showed that 105 of them will continue to go to library. If the critical value is -1.65 what is your decision.
To determine the decision based on the given information, we need to perform a hypothesis test using the critical value.
Step 1: State the hypotheses.
The null hypothesis (H₀): The proportion of students who enjoy doing research in a library is equal to 36%.
The alternative hypothesis (H₁): The proportion of students who enjoy doing research in a library is not equal to 36%.
Step 2: Calculate the test statistic.
The test statistic, also known as the z-score, can be calculated using the formula:
z = (p̂ - p₀) / √(p₀(1-p₀) / n)
where p̂ is the sample proportion, p₀ is the hypothesized proportion, and n is the sample size.
In this case:
p̂ = 105/326 = 0.321
p₀ = 0.36
n = 326
Substituting the values into the formula:
z = (0.321 - 0.36) / √(0.36 * (1-0.36) / 326)
z = -0.039 / √(0.36 * 0.64 / 326)
z = -0.039 / √(0.2304 / 326)
z = -0.039 / √0.00070828
z ≈ -2.675
Step 3: Determine the critical region.
The critical value given in the question is -1.65. For a two-tailed test, we consider both the left and right critical regions.
Since -2.675 < -1.65, the test statistic falls in the critical region. This means we reject the null hypothesis.
Step 4: State the decision.
Based on the test statistic falling in the critical region, we reject the null hypothesis.