A voltage measurement consists of the sum of a constant...

A voltage measurement consists of the sum of a constant unknown voltage and a Gauss-ian-distributed noise voltage of zero mean and variance $10 \mu V^2$. Thirty independent measurements are made and a sample mean of $100 \mu \mathrm{~V}$ is obtained. Find the corresponding 95\% confidence interval.

Key Concepts

Z-score and Critical Values

When the population variance is known and the data are normally distributed, the z-score is used to determine how many standard errors a value is away from the mean. Critical values from the standard normal distribution (such as 1.96 for a 95% confidence level) are used to calculate the width of the confidence interval around the sample mean.

Sample Mean and Standard Error

The sample mean is the average of a set of observations and serves as an estimator of the true population mean. Its accuracy is quantified by the standard error, which is the standard deviation of the sample mean's distribution. The standard error combines the inherent variability of the individual observations and the number of measurements, shrinking as the sample size increases.

Gaussian Distribution

The Gaussian distribution, also known as the normal distribution, is a probability distribution that is symmetric about its mean and characterized by its mean and variance. It is fundamental in statistics because many errors, such as noise in measurements, tend to follow this distribution due to the many small, independent effects that contribute to the variability.

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true value of an unknown parameter, such as a mean. It is constructed so that there is a specified probability (e.g., 95%) that the interval computed from a random sample contains the true parameter value.

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