Question

Responses to essay questions should be given in complete sentences, using proper grammar and punctuation.

A note about probability statement notation:

<: less than

>: greater than

<=: less than or equal to

>=: greater than or equal to

The most famous geyser in the world, Old faithful in Yellowstone national park, has a mean time between eruptions of 85 minutes. Assume the interval of time between eruptions is normally distributed with a standard deviation of 21.25 minutes. Define the random variable X to be the length of time between eruptions.

Use xbar to denote the distribution of the mean length of time between eruptions.

Question 5) What is the probability that a random sample of 30-time intervals between eruptions has a mean longer than 95 minutes?
a)Give the probability statement with the random variable:
B)State the z-score, rounded to 2 decimal places:
c) Give the answer, rounded to 4 decimal places

Question 6) What effect does increasing the sample size have on the probability?

Question 7) What might you conclude if a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?

Question 8) On a certain day, suppose there are 22 time intervals for Old Faithful. Treating these 22 eruptions as a random sample, complete the following sentence:

The likelihood the mean length of time between eruptions exceeds _____________ minutes is 0.20.

Give answer rounded to 2 decimal places.

Answer 1

Since it is given that population is normally distributed so using CLT the sampling distribution of sample mean will be approximately normal with mean and SD as follows: H = !! = 85 = = Vn 21.25/sqrt( 30) = 3.8797

5:

The Z-score for i=95 is: K- os (95 - 85 )/3.8797 = 2.58 Using z table, the probability that I is greater than 95 is: PC I > 95 ) = 1 - Pl I <=95) =1 - P(Z <= 2.58 ) = 1 -0.9951 = 0.0049 Answer: 0.0049 Excel can be used to find the probabilites is =ROUND(1-NORMSDIST(2.58), 4).

6: As the sample size increases, z-score will also increased. So probability of that intervals between eruptions has a mean longer than 95 minute for give random sample will decrease.

7:

Since the probability that a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes is only 0.0049 so it is an unusual event.

8:

Here we need to find eruption time that has 0.20 area to its right.

Since sample size is greater than equal to 30 so using CLT the sampling distribution of sample mean will be approximately normal with mean and SD as follows: U = = 85 = OF = = 21.25/sqrt(22) = 4.5305

We need to find z-score that has 0.2 area to its right and 0.8 area to its left. Using z table, the z-score 0.84 has approximately 0.8 to its left. The required value that has 0.2 area to its right is: ñ = 2:02 + Ile = (0.84 * 4.5305 ) +85 = 88.81 Hence, the required value is 88.81. Excel can be used to find the Z-score is =ROUND(NORMSINV(1 -0.2), 2).

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