According to scientists, the cockroach has had 300 million...

According to scientists, the cockroach has had 300 million years to develop a resistance to destruction. In a study conducted by researchers for S. C. Johnson \& Son, Inc. (manufacturers of Raid $^{\mathbb{B}}$ and Off $\left.^{(\mathbb{B}}\right)$, 5,000 roaches (the expected number in a roach-infested house) were released in the Raid test kitchen. One week later, the kitchen was fumigated and 16,298 dead roaches were counted, a gain of 11,298 roaches for the 1 -week period. Assume that none of the original roaches died during the 1 -week period and that the standard deviation of $x,$ the number of roaches produced per roach in a 1 -week period, is $1.5 .$ Use the number of roaches produced by the sample of 5,000 roaches to find a $95 \%$ confidence interval for the mean number of roaches produced per week for each roach in a typical roach-infested house.

Key Concepts

Central Limit Theorem

The Central Limit Theorem states that, given a sufficiently large sample size, the sampling distribution of the sample mean will approximate a normal distribution regardless of the original population's distribution. This theorem is fundamental in justifying the construction of confidence intervals using the normal (Z) distribution.

Z-Distribution (Normal Distribution)

When the population standard deviation is known or the sample size is large enough, the Z-distribution (or normal distribution) is used to obtain critical values for hypothesis testing and to construct confidence intervals. The use of a Z-score quantifies how many standard errors a value is away from the mean.

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, most often the mean. It quantifies the expected variability of the sample mean around the true population mean and is calculated by dividing the known or estimated population standard deviation by the square root of the sample size.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter (such as the mean) with a certain level of confidence (commonly 95%). It provides an estimate of uncertainty and reliability about the sample-derived estimate for the parameter.

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and a sample estimate. It is typically computed by multiplying the standard error by a critical value associated with the desired confidence level (for example, the Z-score for a 95% confidence interval).

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