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 make a decision based on the given information, we need to conduct a hypothesis test using the critical value and the sample data.
Here, we are trying to assess whether the proportion of students who still enjoy doing research in a library is significantly different from 36% (the population proportion). We can set up the following null and alternative hypotheses:
Null Hypothesis (H0): The proportion of students who still enjoy doing research in a library is equal to 36%.
Alternative Hypothesis (Ha): The proportion of students who still enjoy doing research in a library is different from 36%.
Now, let's calculate the test statistic, which is the z-score in this case. The formula to calculate the z-score for testing a proportion is:
z = (p̂ - p) / √(p * (1-p) / n)
Where:
p̂ is the sample proportion (number of students continuing to go to the library / sample size)
p is the population proportion (36% or 0.36)
n is the sample size (326 in this case)
Calculating the sample proportion:
p̂ = Number of students continuing to go to the library / Sample size
p̂ = 105 / 326
p̂ ≈ 0.322
Calculating the standard error:
SE = √(p * (1 - p) / n)
SE = √(0.36 * 0.64 / 326)
SE ≈ 0.023
Now, let's calculate the z-score:
z = (p̂ - p) / SE
z = (0.322 - 0.36) / 0.023
z ≈ -1.65
The critical value given is -1.65. To make a decision, we compare the calculated z-score with the critical value.
In this case, the calculated z-score (-1.65) is less than the critical value (-1.65). Therefore, the calculated z-score falls within the critical region. We can reject the null hypothesis.
Conclusion: Based on the given sample data and the critical value, we can conclude that there is evidence to suggest that the proportion of students who still enjoy doing research in a library is different from 36%.