In: Statistics and Probability
What price do farmers get for their watermelon crops? In the third week of July, a random sample of 41 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $1.90 per 100 pounds.
(a) 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 | $ |
Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient's total calcium tests gave the following readings (in mg/dl). Assume that the population of x values has an approximately normal distribution.
9.5 | 8.8 | 10.5 | 9.1 | 9.4 | 9.8 | 10.0 | 9.9 | 11.2 |
12.1 |
(b) Find a 99.9% confidence interval for the population mean of total calcium in this patient's blood. (Round your answer to two decimal places.)
lower limit | mg/dl |
upper limit | mg/dl |
What percentage of hospitals provide at least some charity care? Based on a random sample of hospital reports from eastern states, the following information is obtained (units in percentage of hospitals providing at least some charity care):
57.1 | 56 | 53.1 | 66 | 59.0 | 64.7 | 70.1 | 64.7 | 53.5 | 78.2 |
(c) Find a 90% confidence interval for the population average μ of the percentage of hospitals providing at least some charity care. (Round your answers to one decimal place.)
lower limit | % |
upper limit | % |
a)
population std dev , σ = 1.9000
Sample Size , n = 41
Sample Mean, x̅ = 6.8800
Level of Significance , α =
0.1
' ' '
z value= z α/2= 1.6449 [Excel
formula =NORMSINV(α/2) ]
Standard Error , SE = σ/√n = 1.900 /
√ 41 = 0.2967
margin of error, E=Z*SE = 1.6449
* 0.297 = 0.488
confidence interval is
Interval Lower Limit = x̅ - E = 6.88
- 0.488 = 6.3919
Interval Upper Limit = x̅ + E = 6.88
- 0.488 = 7.3681
now, 1 ton=2000 pounds
15 ton = 30000 pounds=300 hundred pounds
so, margin of error = 0.488*300=146.42
lower limit=6.3919*300=1917.58
upper limit=7.3681*300=2210.42
b)
sample std dev , s = √(Σ(X- x̅ )²/(n-1) )
= 1.0001
Sample Size , n = 10
Sample Mean, x̅ = ΣX/n =
10.0300
Level of Significance , α =
0.001
degree of freedom= DF=n-1= 9
't value=' tα/2= 4.7809 [Excel
formula =t.inv(α/2,df) ]
Standard Error , SE = s/√n = 1.000 /
√ 10 = 0.3162
margin of error , E=t*SE = 4.7809
* 0.316 = 1.512
confidence interval is
Interval Lower Limit = x̅ - E = 10.03
- 1.512 = 8.5181
Interval Upper Limit = x̅ + E = 10.03
- 1.512 = 11.5419
99.9% confidence interval is (
8.52 < µ < 11.54
)
c)
sample std dev , s = √(Σ(X- x̅ )²/(n-1) )
= 8.0175
Sample Size , n = 10
Sample Mean, x̅ = ΣX/n =
62.2400
Level of Significance , α =
0.1
degree of freedom= DF=n-1= 9
't value=' tα/2= 1.8331 [Excel
formula =t.inv(α/2,df) ]
Standard Error , SE = s/√n = 8.018 /
√ 10 = 2.5354
margin of error , E=t*SE = 1.8331
* 2.535 = 4.648
confidence interval is
Interval Lower Limit = x̅ - E = 62.24
- 4.648 = 57.5924
Interval Upper Limit = x̅ + E = 62.24
- 4.648 = 66.8876
90% confidence interval is ( 57.6
< µ < 66.9 )