Question

 

The state of California has a mean annual rainfall of 22 inches, whereas the state of New York has a mean annual rainfall of 42 inches. Assume that the standard deviation for both states is 4 inches. A sample of 34 years of rainfall for California and a sample of 46 years of rainfall for New York has been taken.

(a)

Show the probability distribution of the sample mean annual rainfall for California.

A bell-shaped curve is above a horizontal axis labeled inches.

• The horizontal axis ranges from about −2.1 to about 2.1.
• The curve enters the viewing window near −2.1 just above the horizontal axis, curves up to the right, and reaches a maximum near 0.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 2.1.

A bell-shaped curve is above a horizontal axis labeled inches.

• The horizontal axis ranges from about 39.9 to about 44.1.
• The curve enters the viewing window near 39.9 just above the horizontal axis, curves up to the right, and reaches a maximum near 42.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 44.1.

A bell-shaped curve is above a horizontal axis labeled inches.

• The horizontal axis ranges from about 19.9 to about 24.1.
• The curve enters the viewing window near 19.9 just above the horizontal axis, curves up to the right, and reaches a maximum near 22.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 24.1.

A bell-shaped curve is above a horizontal axis labeled inches.

• The horizontal axis ranges from about 10 to about 34.
• The curve enters the viewing window near 10 just above the horizontal axis, curves up to the right, and reaches a maximum near 22.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 34.

(b)

What is the probability that the sample mean is within 1 inch of the population mean for California? (Round your answer to four decimal places.)

(c)

What is the probability that the sample mean is within 1 inch of the population mean for New York? (Round your answer to four decimal places.)

(d)

In which case, part (b) or part (c), is the probability of obtaining a sample mean within 1 inch of the population mean greater? Why?

part (c), because the population standard deviation is smallerpart (c), because the sample size is larger    part (b), because the standard error is smallerpart (b), because the population standard deviation is smaller

Answer 1

(a)

The probability distribution of the sample mean annual rainfall for California.

mean annual rainfall = 22 inches

Standard deviation of mean annual rainfall = 4 / = 0.686

Lower range = 22 - 3 * 0.686   19.9

Upper range = 22 + 3 * 0.686   24.1

A bell-shaped curve is above a horizontal axis labeled inches.

• The horizontal axis ranges from about 19.9 to about 24.1.
• The curve enters the viewing window near 19.9 just above the horizontal axis, curves up to the right, and reaches a maximum near 22.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 24.1.

(b)

What is the probability that the sample mean is within 1 inch of the population mean for California? (Round your answer to four decimal places.)

P(22-1 < < 22+1) = P( < 23) - P( < 21)

= P[Z < (23 - 22)/0.686] - P[Z < (21 - 22)/0.686]

= P[Z < 1.46] - P[Z < -1.46]

= 0.9279 - 0.0721

= 0.8558

(c)

What is the probability that the sample mean is within 1 inch of the population mean for New York? (Round your answer to four decimal places.)

Standard deviation of mean annual rainfall = 4 / = 0.5898

P(42-1 < < 42+1) = P( < 43) - P( < 41)

= P[Z < (43 - 42)/0.5898] - P[Z < (41 - 42)/0.5898]

= P[Z < 1.70] - P[Z < -1.70]

= 0.9554 - 0.0446

= 0.9108

(d)

In which case, part (b) or part (c), is the probability of obtaining a sample mean within 1 inch of the population mean greater? Why?

part (c), because the sample size is larger   

which reduces the standard error of mean rainfall for NewYork