Among A Group Of 165 Students, 8 Are Taking Calculus Discret

Among A Group Of 165 Students 8 Are Taking Calculus Discrete Structur

Among a group of 165 students, various subsets are enrolled in different subjects: 8 students are taking calculus, discrete structures, and computer science; 33 students are taking calculus and computer science; 20 students are taking calculus and discrete structures; 24 students are taking discrete structures and computer science; 79 students are taking calculus; 83 students are taking discrete structures; and 63 students are taking computer science. The task is to determine how many students are not enrolled in any of these three subjects.

To solve this problem, we employ principles from set theory, specifically the inclusion-exclusion principle, which allows us to handle overlaps among multiple sets efficiently.

Applying the Inclusion-Exclusion Principle

Let:

• C = set of students taking calculus
• D = set of students taking discrete structures
• S = set of students taking computer science

Given data:

• n(C) = 79
• n(D) = 83
• n(S) = 63
• n(C ∩ D) = 20
• n(C ∩ S) = 33
• n(D ∩ S) = 24
• n(C ∩ D ∩ S) = 8

Using the inclusion-exclusion formula for three sets, the total number of students enrolled in at least one subject is:

N(C ∪ D ∪ S) = n(C) + n(D) + n(S) - n(C ∩ D) - n(C ∩ S) - n(D ∩ S) + n(C ∩ D ∩ S)

Substituting the known values:

N(C ∪ D ∪ S) = 79 + 83 + 63 - 20 - 33 - 24 + 8 = 205 - 77 + 8 = 136

Since the total number of students in the group is 165, the number of students not enrolled in any of the three subjects is:

Number of students with none = 165 - 136 = 29

Therefore, 29 students are not enrolled in any of these subjects.

Estimating the Number of Women with Nickel Allergy in the Study

The second part of the question pertains to the estimation of the population proportion with a nickel allergy based on two samples of women with dermatitis caused by eye shadow or mascara. We are to determine the minimum sample size needed in each group to estimate the true population proportion within a margin of error of 3% (0.03), with a certain level of confidence. The data provided is:

• Group 1 (eye shadow): n₁ = 131 women, with 12 diagnosed with nickel allergy
• Group 2 (mascara): n₂ = 250 women, with 25 diagnosed with nickel allergy

Estimation of Sample Size for Proportion

To achieve a specified margin of error (E) in estimating a population proportion p, the sample size n can be calculated using the formula:

n = (Z_α/2)² * p̂(1 - p̂) / E²

Where:

• Z_α/2 is the critical value for the chosen confidence level (for 95%, Z_α/2 ≈ 1.96)
• p̂ is the estimated proportion from the sample
• E is the margin of error (0.03)

Calculating the Sample Size for Each Group

Group 1 (eye shadow):

The sample proportion p̂₁ is:

p̂₁ = 12 / 131 ≈ 0.0916

Using p̂ ≈ 0.0916, the maximum variability occurs around p̂ = 0.5; however, to be conservative and since p̂ is small, we proceed with the estimated p̂.

Calculating n₁:

n₁ = (1.96)^2 0.0916 (1 - 0.0916) / (0.03)^2 ≈ 3.8416 0.0916 0.9084 / 0.0009 ≈ 3.8416 * 0.0833 / 0.0009 ≈ 0.3203 / 0.0009 ≈ 356

Group 2 (mascara):

The sample proportion p̂₂ is:

p̂₂ = 25 / 250 = 0.10

Calculating n₂:

n₂ = 1.96^2 0.10 0.90 / 0.0009 ≈ 3.8416 * 0.09 / 0.0009 ≈ 0.3457 / 0.0009 ≈ 384

Interpretation and Conclusion

Based on these calculations, to estimate the true proportion of women with nickel allergy within 3% margin of error with 95% confidence, approximately 356 women in the eye shadow group and 384 women in the mascara group need to be sampled. The actual sample sizes used in the study (131 and 250) are smaller, which means the actual margin of error would be larger. Increasing the sample size improves the precision of the estimate.

Implications for Future Research

The analysis highlights the importance of adequate sample sizes in epidemiological studies to obtain reliable estimates. Larger samples reduce the margin of error and lead to more precise estimates of the true population parameter. Researchers should plan sample sizes based on preliminary data to ensure they meet their desired confidence and precision levels.

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