The most famous geyser in the world, Old Faithful in...

The most famous geyser in the world, Old Faithful in Yellowstone National Park, has a mean time between eruptions of 85 minutes. If the interval of time between eruptions is normally distributed with standard deviation 21.25 minutes, answer the following questions: Source: www.unmuseum.org (a) What is the probability that a randomly selected time interval between eruptions is longer than 95 minutes? (b) What is the probability that a random sample of 20 time intervals between eruptions has a mean longer than 95 minutes? (c) What is the probability that a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes? (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. (e) What might you conclude if a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes? (f) On a certain day, suppose there are 22 time intervals for Old Faithful. Treating these 22 eruptions as a random sample, the likelihood the mean length of time between eruptions exceeds __________ minutes is 0.20.

The most famous geyser in the world, Old Faithful in Yellowstone National Park, has a mean time between eruptions of 85 minutes. If the interval of time between eruptions is normally distributed with standard deviation 21.25 minutes, answer the following questions:
Source: www.unmuseum.org
(a) What is the probability that a randomly selected time interval between eruptions is longer than 95 minutes?
(b) What is the probability that a random sample of 20 time intervals between eruptions has a mean longer than 95 minutes?
(c) What is the probability that a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?
(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result.
(e) What might you conclude if a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?
(f) On a certain day, suppose there are 22 time intervals for Old Faithful. Treating these 22 eruptions as a random sample, the likelihood the mean length of time between eruptions exceeds __________ minutes is 0.20.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is fundamental in statistics, as many natural phenomena tend to be normally distributed, and it provides the basis for many inference procedures. In the context of the problem, the time between eruptions follows a normal distribution, which allows for the utilization of z-scores to find probabilities for specific time intervals.

Z-Score

A z-score is a statistical measurement that describes a value’s position relative to the mean of a group of values, measured in terms of standard deviations. It is used to standardize scores from a normal distribution so that probabilities can be calculated using the standard normal distribution table. In the problem, z-scores are calculated to determine the probability that a given time interval exceeds a certain threshold.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible sample means of a given size drawn from the population. It is central to inferential statistics and is particularly important because of the Central Limit Theorem. In the problem, when forming random samples of different sizes, the distribution of the sample means is used to calculate probabilities associated with observed averages.

Central Limit Theorem

The Central Limit Theorem (CLT) states that when a large enough sample size is drawn from a population with any distribution, the sample mean will be approximately normally distributed, regardless of the original population's distribution. This theorem is essential in the problem because it justifies using normal probability calculations for the sample means, even when the criteria of individual measurements differ.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean and indicates how much the sample mean is expected to vary from the true population mean. It is calculated as the population standard deviation divided by the square root of the sample size. A larger sample size results in a smaller standard error, making the estimate of the mean more precise. This concept explains why increasing the sample size in the problem affects the probability associated with the sample mean.

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