A manufacturer knows that the numbers of items produced per...

A manufacturer knows that the numbers of items produced per hour by machine A and by machine B are normally distributed with a standard deviation of $8.4$ items for machine A and a standard deviation of $11.3$ items for machine B. The mean hourly amount produced by machine A for a random sample of 40 hours was 130 units; the mean hourly amount produced by machine B for a random sample of 36 hours was 120 units. Find the $95 \%$ confidence interval for the difference in mean parts produced per hour by these two machines.

Key Concepts

Z-Statistic

When population standard deviations are known and the sample size is sufficiently large, the z-statistic can be used to perform hypothesis tests and construct confidence intervals. It represents the number of standard deviations a data point or statistic is from the mean of the distribution and relies on the properties of the standard normal distribution.

Standard Error

The standard error is a measure of the variability of a sample statistic, such as the mean, and it quantifies the precision of the sample estimate of the population parameter. It is calculated from the standard deviation and the sample size, with larger samples yielding smaller standard errors.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Its bell-shaped curve is fundamental in statistics, especially since many real-world variables tend to approximate this distribution under certain conditions.

Sampling Distribution of the Sample Mean

This concept refers to the distribution of the sample mean taken over many samples from the same population. According to the central limit theorem, regardless of the shape of the underlying population distribution, the distribution of the sample mean approximates a normal distribution as the sample size increases.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. It quantifies the uncertainty associated with a sample estimate and is usually expressed with a confidence level (e.g., 95%), which indicates the probability that the interval includes the true parameter.

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