Question

1) Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma.† Over a period of months, an adult male patient has taken eight blood tests for uric acid. The mean concentration was x = 5.35 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with σ = 1.93 mg/dl.

(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient's blood. What is the margin of error? (Round your answers to two decimal places.)

lower limit
upper limit
margin of error

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.06 for the mean concentration of uric acid in this patient's blood. (Round your answer up to the nearest whole number.)

2) Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 10 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.34 gram.

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit
upper limit
margin of error

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.08 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

3) How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds, by Wirth and Young (Random House), claims that the air in the crown should be an average of 100°C for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly 100°C. What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 52 readings (for a balloon in equilibrium) gave a mean temperature of x = 97°C. For this balloon, σ ≈18°C.

(a) Compute a 95% confidence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (Round your answers to one decimal place.)

lower limit	°C
upper limit

Answer 1

Solution:

Question1)

Given:

n = sample size = an adult male patient has taken eight blood tests for uric acid = 8

Sample mean =

The distribution of uric acid in healthy adult males can be assumed to be normal, with σ = 1.93 mg/dl.

Part a) We have to find a 95% confidence interval for the population mean concentration of uric acid in this patient's blood and margin of error.

Since population is normally distributed and population standard deviation σ is known, we use z ( Standard Normal) distribution to find confidence interval.

Formula is:

where

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus margin of Error is:

Lower limit =

Upper Limit =

Thus

Lower Limit = 4.01

Upper Limit = 6.69

Margin of Error = E = 1.34

Part d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.06 for the mean concentration of uric acid in this patient's blood.

Formula for sample size n is:

Thus required sample size is n = 13.