Question

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 the eruptions is normally distributed with standard deviation 21.25 minutes, complete parts (a) through (f). click here to view the standard normal distribution table (page 1). click here to view the standard normal distribution table (page 2). (a) what is the probability that a randomly selected time interval between eruptions is longer than 94 minutes? the probability that a randomly selected time interval is longer than 94 minutes is approximately 0.3372. (b) what is the probability that a random sample of 10 time intervals between eruptions has a mean longer than 94 minutes? the probability that the mean of a random sample of 10 time intervals is more than 94 minutes is approximately 0.0601 (round to four decimal places as needed.) (c) what is the probability that a random sample of 34 time intervals between eruptions has a mean longer than 94 minutes? the probability that the mean of a random sample of 34 time intervals is more than 94 minutes is approximately 0.0068 (round to four decimal places as needed.) (d) what effect does increasing the sample size have on the probability? provide an explanation for this result. if the sample size increases, the probability because the variability in the sample mean

Explanation:

Step1: Recall the concept of sampling distribution

The standard deviation of the sample mean (standard error) is given by $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.

Step2: Analyze the effect of increasing $n$

As the sample size $n$ increases, the value of $\sqrt{n}$ increases, so the standard error $\sigma_{\bar{x}}$ decreases. A smaller standard error means that the sample means are more concentrated around the population mean, reducing the variability in the sample mean.

Step3: Relate to probability

For a given value (here, a sample mean greater than 94 minutes), a smaller standard error (less variability) means that the area in the tail of the sampling distribution (the probability that the sample mean is in that tail) will decrease. So increasing the sample size decreases the probability (for the event of the sample mean being more than 94 minutes) because the variability in the sample mean decreases.

Answer:

decreases; decreases; the standard error (or variability) of the sample mean decreases as sample size increases.