Question

Help: A random sample of n measurements was selected from a population with an unknown mean and standard deviation

σ=30 for each of the situations in parts a through d. Calculate a 95% confidence interval for each of these situations.

a. n=75, X=25

b. n=250, X=116

c. n=90, X=16

d. n=90, X=5.21 Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a through d? Explain.
a. (20.28,29.72)

b. (112.43,119.57)

c. (12.53,19.47)

d. (3.69,6.73)

Answer 1

To calculate a 95% confidence interval for a known population standard deviation, use the population standard deviation (σ), the sample mean (x), and the Z-score for 95% confidence, which is typically 1.96. The assumption of normality is not strictly necessary for large samples due to the Central Limit Theorem, but it is important for smaller or skewed distributions.

Step-by-step explanation:

To compute a 95% confidence interval for a population mean when the population standard deviation (σ) is known, we use the following formula:

CI = (x ± (Z*σ/√n))

where:

• x is the sample mean.
• Z is the Z-score associated with the desired confidence level.
• σ is the population standard deviation.
• n is the sample size.

For a 95% confidence level, the Z-score is typically 1.96. Substituting values for each situation gives us:

1. n=75, x=25, σ=30: CI = (20.28, 29.72)
2. n=250, x=116, σ=30: CI = (112.43, 119.57)
3. n=90, x=16, σ=30: CI = (12.53, 19.47)
4. n=90, x=5.21, σ=30: CI = (3.69, 6.73)

The assumption that the underlying population is normally distributed is not strictly necessary when the sample size is large (typically n ≥ 30) due to the Central Limit Theorem, which states that the distribution of the sample means will be approximately normal, regardless of the population's distribution. However, for small sample sizes or skewed distributions, this assumption is important to ensure the validity of the confidence intervals.