In the past a certain college has found that 35% of the students will grad within 4 years. A random sample of 450 students at this college is taken and going to be followed until graduation.
A. What is the sampling distribution of the statistic of interest?
B. What is the probality that 135 or less students grad within 4 years for the sample of 450 students??
C. Based on your probability in part b, what do you think about the claim 35% of students will grad in 4 years at this college
A. The sampling distribution of the statistic of interest, which is the proportion of students who will graduate within 4 years, can be approximated as a normal distribution. The mean of this distribution would be equal to the population proportion (35%) and the standard deviation can be calculated using the formula:
Standard Deviation = √[(p * (1 - p)) / n]
Where p is the population proportion and n is the sample size.
B. To calculate the probability that 135 or fewer students graduate within 4 years for the sample of 450 students, we need to use the normal distribution. We can calculate the z-score for the value of 135 using the formula:
z = (x - μ) / σ
Where x is the observed value, μ is the mean, and σ is the standard deviation.
Once we have the z-score, we can use a standard normal table or calculator to find the corresponding probability.
C. Based on the probability calculated in part B, we can make inference about the claim that 35% of students will graduate in 4 years at this college. If the probability obtained in part B is significantly lower than a certain threshold (usually set at 0.05), we can reject the claim and conclude that the actual proportion of students graduating within 4 years is different from 35%. However, if the probability is not significantly lower, we cannot reject the claim and conclude that there is not enough evidence to suggest that the actual proportion is different from 35%.
A. The sampling distribution of the statistic of interest, which is the proportion of students who graduate within 4 years, can be approximated by a normal distribution. This approximation holds true when the sample size is sufficiently large (n ≥ 30) and the population proportion is not extremely close to 0 or 1.
B. To find the probability that 135 or fewer students graduate within 4 years in a sample of 450 students, we need to use the normal approximation. First, we calculate the sample proportion:
Sample Proportion (p̂) = Number of students who graduate within 4 years / Total sample size
= 135 / 450
= 0.3
Next, we need to calculate the standard deviation (σ) of the sampling distribution, which is also called the standard error:
Standard Error (SE) = sqrt[(p̂ * (1 - p̂)) / n]
= sqrt[(0.3 * (1 - 0.3)) / 450]
≈ 0.0183 (rounded to 4 decimal places)
Now, we can use the normal distribution to find the probability. We need to calculate the z-score:
z = (x - μ) / σ
x = 135 (number of students who graduate within 4 years)
μ = n * p̂ = 450 * 0.35 = 157.5 (expected value or mean)
σ = standard error = 0.0183
z = (135 - 157.5) / 0.0183
≈ -123.28
Using a standard normal distribution table or a calculator, we find the probability P(z ≤ -123.28) is essentially 0.
C. Based on the probability calculated in part B, we can conclude that it is highly unlikely for only 135 or fewer students to graduate within 4 years in a random sample of 450 students. This suggests that the claim of 35% of students graduating within 4 years at this college may not be accurate, as the observed sample proportion is significantly lower. However, further statistical tests and analysis would be required to make a definitive conclusion.