Question

What price do farmers get for their watermelon crops? In the third week of July, a random sample of 42 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $2.00 per 100 pounds.

(a) Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the margin of error? (Round your answers to two decimal places.)

lower limit	$
upper limit	$
margin of error	$

(b) Find the sample size necessary for a 90% confidence level with maximal error of estimate E = 0.29 for the mean price per 100 pounds of watermelon. (Round up to the nearest whole number.)
farming regions

(c) A farm brings 15 tons of watermelon to market. Find a 90% confidence interval for the population mean cash value of this crop. What is the margin of error? Hint: 1 ton is 2000 pounds. (Round your answers to two decimal places.)

lower limit	$
upper limit	$
margin of error	$

At wind speeds above 1000 centimeters per second (cm/sec), significant sand-moving events begin to occur. Wind speeds below 1000 cm/sec deposit sand and wind speeds above 1000 cm/sec move sand to new locations. The cyclic nature of wind and moving sand determines the shape and location of large dunes. At a test site, the prevailing direction of the wind did not change noticeably. However, the velocity did change. Sixty-five wind speed readings gave an average velocity of x = 1075 cm/sec. Based on long-term experience, σ can be assumed to be 275 cm/sec.

(a) Find a 95% confidence interval for the population mean wind speed at this site. (Round your answers to the nearest whole number.)

lower limit	cm/sec
upper limit	cm/sec

(b) Does the confidence interval indicate that the population mean wind speed is such that the sand is always moving at this site? Explain.

No. This interval indicates that the population mean wind speed is such that the sand may not always be moving at this site.Yes. This interval indicates that the population mean wind speed is such that the sand may not always be moving at this site.     Yes. This interval indicates that the population mean wind speed is such that the sand is always moving at this site.No. This interval indicates that the population mean wind speed is such that the sand is always moving at this site.

Answer 1

(a)

sample mean of x = $6.88 per 100 pounds

standard error of mean = σ / = 2 / = 0.3086

For 90% confidence interval, z = 1.645

Margin of error = z * Std error = 1.645 * 0.3086 = $0.5076

lower limit = 6.88 - 0.5076 = $6.3724

upper limit = 6.88 + 0.5076 = $7.3876

(b)

E = 0.29

Sample size, n = (z σ / E)²

= (1.645 * 2 / 0.29)²

= 129 (Rounded to nearest integer)

(c)

For 15 tons of watermelon, population mean = (6.88 / 100) * 15 * 2000 = 2064

σ = (2 / 100) * 15 * 2000 = 600

standard error of mean = σ / = 600 / = 92.58201

For 90% confidence interval, z = 1.645

Margin of error = z * Std error = 1.645 * 92.58201 = $152.2974

lower limit = 2064 - 152.2974 = $1911.703

upper limit = 2064 + 152.2974 = $2216.297

(a)

Sample mean = x = 1075 cm/sec

σ = 275

standard error of mean = σ / = 275 / = 34.10955

For 95% confidence interval, z = 1.96

Margin of error = z * Std error = 1.96 * 34.10955 = $66.8547

lower limit = 1075 - 66.8547 = $1008.145

upper limit = 1075 + 66.8547 = $1141.855

(b)

Since the lower limit is greater than 1000 centimeters per second, the answer is,

Yes. This interval indicates that the population mean wind speed is such that the sand is always moving at this site