A random sample of n measurements was selected from EwV a...

A random sample of $n$ measurements was selected from EwV a population with unknown mean $\mu$ and known standard deviation $\sigma$. Calculate a $95 \%$ confidence interval for $\mu$ for each of the following situations: a. $n=75, \bar{x}=28, \sigma^2=12$ b. $n=200, \bar{x}=102, \sigma^2=22$ c. $n=100, \bar{x}=15, \sigma=.3$ d. $n=100, \bar{x}=4.05, \sigma=.83$ e. Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a-d? Explain.

A random sample of $n$ measurements was selected from EwV a population with unknown mean $\mu$ and known standard deviation $\sigma$. Calculate a $95 \%$ confidence interval for $\mu$ for each of the following situations:
a. $n=75, \bar{x}=28, \sigma^2=12$
b. $n=200, \bar{x}=102, \sigma^2=22$
c. $n=100, \bar{x}=15, \sigma=.3$
d. $n=100, \bar{x}=4.05, \sigma=.83$
e. Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a-d? Explain.

Key Concepts

Confidence Interval

A confidence interval is a range of values derived from sample statistics that is likely to contain the population parameter, in this case the mean (?), with a certain level of confidence (95% in this instance). It incorporates both the sample estimate and the uncertainty due to sampling variability.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean, calculated by dividing the known population standard deviation by the square root of the sample size. It measures how much the sample mean is expected to vary from one sample to another.

Z-Distribution and Critical Values

When the population standard deviation is known, the z-distribution is used to determine the critical values for the confidence interval. The 95% confidence level corresponds to specific z-scores (approximately ±1.96) which define the margin of error around the sample mean.

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution. This justifies the use of the normal (z) distribution for calculating confidence intervals, even when the underlying population distribution is not normal for large samples.

Assumptions About the Underlying Population

For small sample sizes, the normality assumption of the underlying population is crucial to ensure the validity of the confidence interval calculated using z-scores. However, for larger samples, the Central Limit Theorem mitigates this requirement as the sampling distribution of the mean becomes approximately normal, even if the original data is not.

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