Measurements on the percentage of enrichment of 12 fuel rods used in a nuclear reactor were reported as follows;
3.11 2.88 3.08 3.01 2.84 2.86 3.04 3.09 3.08 2.89 3.12 2.98
a. Test the Hypothesis H0: µ = 2.95 versus H1: µ 0 ≠ 2.95, and draw appropriate conclusions. Use the P-value approach.
b. Find a 99% two-sided CI on the mean percentage of enrichment. Are you confortable with the statement that the mean percentage of enrichment is 2.95%? Why?
c. What would you use to check the normality assumption of the data?
a. To test the hypothesis H0: µ = 2.95 versus H1: µ ≠ 2.95, we can calculate the sample mean (x̄) and the sample standard deviation (s) from the provided measurements. Then, we can use the t-test to determine the P-value.
1. Calculate the sample mean (x̄):
x̄ = (3.11 + 2.88 + 3.08 + 3.01 + 2.84 + 2.86 + 3.04 + 3.09 + 3.08 + 2.89 + 3.12 + 2.98) / 12 = 2.9742
2. Calculate the sample standard deviation (s):
s = sqrt([(x1 - x̄)^2 + (x2 - x̄)^2 + ... + (xn - x̄)^2] / (n-1))
= sqrt([ (3.11 - 2.9742)^2 + (2.88 - 2.9742)^2 + ... + (2.98 - 2.9742)^2 ] / 11)
= 0.1047
3. Calculate the t-value:
t = (x̄ - µ) / (s / sqrt(n))
= (2.9742 - 2.95) / (0.1047 / sqrt(12))
= 0.6942
4. Determine the degrees of freedom (df) based on the given sample size (n):
df = n - 1
= 12 - 1
= 11
5. Calculate the two-sided P-value using the t-distribution table or a statistical calculator:
P-value = 2 * P(t > |t|)
= 2 * P(t > 0.6942) // since the value is positive
= 2 * (1 - P(t < 0.6942))
6. Compare the calculated P-value to the significance level (α) to draw appropriate conclusions. If the P-value is less than α, we reject the null hypothesis H0. Typically, a significance level of 0.05 (5%) is used.
b. To find a 99% two-sided confidence interval (CI) on the mean percentage of enrichment, we should use a t-distribution. The formula to calculate the CI is:
CI = x̄ ± (t * s / sqrt(n))
Where x̄ is the sample mean, t is the t-value corresponding to the desired confidence level and degrees of freedom, s is the sample standard deviation, and n is the sample size.
c. To check the normality assumption of the data, we can employ various methods like graphical methods (such as histogram, boxplot, or Q-Q plot) or statistical tests (such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test). These methods can help assess the distribution of the data and determine if it approximates a normal distribution.