The Same Study From Question 1 Focused On Women Aged 20–29

The Same Study From Question 1 Focused On Women Aged 20 29 Years T

The study referenced in question 1 focused on women aged 20-29 years. The mean cholesterol level for the entire population in that age group was 183 mg/dl, and the standard deviation was 37 mg/dl. A sample of 60 women was drawn from this population. The assignment involves calculating the probabilities associated with the sample mean serum cholesterol levels, specifically:

a. The probability that the sample mean is between 170 mg/dl and 195 mg/dl.

b. The probability that the sample mean is below 175 mg/dl.

c. The probability that the sample mean is greater than 190 mg/dl.

Additionally, there is a separate question involving binomial probability:

(a) For n=13 and p=0.3, find the binomial probability of p(5) using a binomial probability table.

(b) If np ≥ 5 and nq ≥ 5, estimate the binomial probability using the normal distribution as an approximation. If np

Paper For Above instruction

The analysis of cholesterol levels within a specific demographic provides insights into health risks and aids in clinical decision-making. In this study, which pertains to women aged 20-29 years, the statistical evaluation revolves around the properties of the sample mean and binomial probabilities. The calculations hinge on understanding the distribution of sample means, as well as applying binomial probability techniques and approximations.

First, considering the cholesterol data, the population mean (μ) is 183 mg/dl with a standard deviation (σ) of 37 mg/dl. When a sample of size n=60 is drawn from this population, the sampling distribution of the sample mean (x̄) can be approximated as normal, thanks to the Central Limit Theorem. The mean of this distribution remains μ=183, while its standard error (SE) is σ/√n = 37/√60, approximately 4.77 mg/dl. This standard error measures the variability of the sample mean around the population mean for samples of this size.

To find the probability that the sample mean falls between specific values, the standard normal distribution (z-distribution) is utilized. The z-score is calculated as (x̄ - μ) / SE.

a) Probability that the mean is between 170 mg/dl and 195 mg/dl:

Calculating each z-score:

- For 170 mg/dl: z = (170 - 183) / 4.77 ≈ -2.58

- For 195 mg/dl: z = (195 - 183) / 4.77 ≈ 2.52

Consulting standard normal distribution tables:

- P(z

- P(z

Therefore, the probability that the sample mean is between 170 and 195 mg/dl:

P = P(z

b) Probability that the mean is below 175 mg/dl:

Z-score for 175 mg/dl:

z = (175 - 183) / 4.77 ≈ -1.58

From the z-table:

P(z

Thus, the probability that the sample mean is below 175 mg/dl is approximately 5.7%.

c) Probability that the mean exceeds 190 mg/dl:

Z-score for 190 mg/dl:

z = (190 - 183) / 4.77 ≈ 1.47

Corresponding probability:

P(z > 1.47) = 1 - P(z

Hence, there is roughly a 7.08% chance that the sample mean exceeds 190 mg/dl.

Moving to the binomial probability question, where n=13 and p=0.3:

(a) To find p(5), the probability of exactly 5 successes in 13 trials, the binomial probability formula is used:

P(X=5) = C(13,5) × p^5 × (1-p)^{8}

Calculating C(13,5):

C(13,5) = 1287

Substituting values:

P(5) = 1287 × (0.3)^5 × (0.7)^8

Calculations:

- (0.3)^5 ≈ 0.00243

- (0.7)^8 ≈ 0.05765

Multiplying:

P(5) ≈ 1287 × 0.00243 × 0.05765 ≈ 1287 × 0.000140 ≈ 0.1803

Therefore, the probability of exactly 5 successes is approximately 0.180.

(b) Regarding the normal approximation to the binomial distribution:

Calculate np = 13 × 0.3 = 3.9

Calculate nq = 13 × 0.7 = 9.1

Since np=3.9

It is important to verify conditions for approximation, which typically require np ≥ 5 and nq ≥ 5 for acceptable accuracy. Since these are not met, the exact binomial calculation provides the most reliable result.

In conclusion, statistical methods such as the normal approximation to the binomial distribution are powerful tools but must be used with caution, respecting their assumptions and limitations. The detailed calculations demonstrate how standard statistical formulas can be applied to real-world health data, such as cholesterol levels, providing meaningful insights into population health risks.

References

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