A manufacturer produces piston rings for an automobile...

A manufacturer produces piston rings for an automobile engine. It is known that ring diameter is normally distributed with $\sigma=0.001$ millimeters. A random sample of 15 rings has a mean diameter of $\bar{x}=74.036$ millimeters. (a) Construct a $99 \%$ two-sided confidence interval on the mean piston ring diameter. (b) Construct a $99 \%$ lower-confidence bound on the mean piston ring diameter. Compare the lower bound of this confidence interval with the one in part (a).

A manufacturer produces piston rings for an automobile engine. It is known that ring diameter is normally distributed with $\sigma=0.001$ millimeters. A random sample of 15 rings has a mean diameter of $\bar{x}=74.036$ millimeters.
(a) Construct a $99 \%$ two-sided confidence interval on the mean piston ring diameter.
(b) Construct a $99 \%$ lower-confidence bound on the mean piston ring diameter. Compare the lower bound of this confidence interval with the one in part (a).

Key Concepts

One-sided Confidence Interval

A one-sided confidence interval, such as a lower confidence bound, provides a cutoff point below or above which a population parameter is unlikely to lie with a specified level of confidence. Unlike two-sided intervals that establish both an upper and lower bound, one-sided intervals focus on estimating only one boundary of the parameter, which can be useful in cases where only an excess or deficit is of interest.

Confidence Interval Construction

Confidence intervals provide a range of values within which the true population parameter, such as the mean, is expected to lie with a certain level of confidence. Constructing a confidence interval involves determining a margin of error, which is influenced by the critical value from the appropriate distribution, the standard error, and the desired confidence level.

Standard Error

The standard error is a measure of the variability of the sample mean estimate from sample to sample. It is calculated by dividing the population standard deviation by the square root of the sample size. This measure is central in constructing confidence intervals, as it quantifies the expected deviation of the sample mean from the true population mean.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of the average values computed from all possible samples of a fixed size from a population. When the original data is normally distributed and the sample size is finite or even small, the sample means are also normally distributed, which allows for the use of standard inferential techniques.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, defined by its mean and variance. It is fundamental in statistics because many random variables tend to be normally distributed, especially under the Central Limit Theorem when sampling from a broader population.

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