1)A sample of 17 patients in a hospital had these hemoglobin readings (grams): 112 120 98 55 71 35 99 124 64
150 150 55 100 132 20 70 93
Find a 95% confidence interval for the hemoglobin reading for all the patients in the hospital.

2)A sample of 500 nursing applications included 60 from men. Find the 90% confidence interval for the true proportion of men who applied to the nursing program.

3)Average undergraduate cost for tuition, fees, room, and board for all institutions last year was $19,410. A random sample of costs this year for 40 institutions of higher learning indicated that the sample mean was $22,098, and the sample standard deviation was $6050. At the α=0.01, is there sufficient evidence to conclude that the cost of attendance has increased?

Step-by-step explanation:

1)

Mean = (112 + 120 + 98 + 55 + 71 + 35 + 99 + 124 + 64 + 150 + 150 + 55 + 100 + 132 + 20 + 70 + 93)/17 = 91.1

Standard deviation = √(summation(x - mean)/n

Summation(x - mean) = (112 - 91.1)^2 + (120 - 91.1)^2 + (98 - 91.1)^2 + (55 - 91.1)^2 + (71 - 91.1)^2 + (35 - 91.1)^2 + (99 - 91.1)^2 + (124 - 91.1)^2 + (64 - 91.1)^2 + (150 - 91.1)^2 + (150 - 91.1)^2 + (55 - 91.1)^2 + (100 - 91.1)^2 + (132 - 91.1)^2 + (20 - 91.1)^2 + (70 - 91.1)^2 + (93 - 91.1)^2 = 23550.97

s = √23550.97/17 = 37.2

Confidence interval is written in the form,

(Sample mean - margin of error, sample mean + margin of error)

The sample mean, x is the point estimate for the population mean.

Margin of error = z × s/√n

Where

s = sample standard deviation = 37.2

n = number of samples = 17

From the information given, the population standard deviation is unknown and the sample size is small, hence, we would use the t distribution to find the z score

In order to use the t distribution, we would determine the degree of freedom, df for the sample.

df = n - 1 = 17 - 1 = 16

Since confidence level = 95% = 0.95, α = 1 - CL = 1 – 0.95 = 0.05

α/2 = 0.05/2 = 0.025

the area to the right of z0.025 is 0.025 and the area to the left of z0.025 is 1 - 0.025 = 0.975

Looking at the t distribution table,

z = 2.12

Margin of error = 2.12 × 37.2/√17

= 19.13

The 95% confidence interval for the hemoglobin reading for all the patients in the hospital is

91.1 ± 19.13

2) Confidence interval for population proportion is written as

Sample proportion ± margin of error

Margin of error = z × √pq/n

Where

z represents the z score corresponding to the confidence level

p = sample proportion. It also means probability of success

q = probability of failure

q = 1 - p

p = x/n

Where

n represents the number of samples

x represents the number of success

From the information given,

n = 500

x = 60

p = 60/500 = 0.12

q = 1 - 0.12 = 0.88

To determine the z score, we subtract the confidence level from 100% to get α

α = 1 - 0.9 = 0.1

α/2 = 0.1/2 = 0.05

This is the area in each tail. Since we want the area in the middle, it becomes

1 - 0.05 = 0.95

The z score corresponding to the area on the z table is 1.645. Thus, Tthe z score for a confidence level of 90% is 1.645

Therefore, the 90% confidence interval is

0.12 ± 1.645√(0.12)(0.88)/500

Confidence interval = 0.12 ± 0.024

3) We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean

For the null hypothesis,

µ = 19410

For the alternative hypothesis,

µ > 19410

This is a right tailed test.

Since the population standard deviation is given, z score would be determined from the normal distribution table. The formula is

z = (x - µ)/(σ/√n)

Where

x = sample mean

µ = population mean

σ = population standard deviation

n = number of samples

From the information given,

µ = 19410

x = 22098

σ = 6050

n = 40

z = (19410 - 22098)/(6050/√40) = - 2.81

Looking at the normal distribution table, the probability corresponding to the z score is 0.0025

p value = 0.0025

Since alpha, 0.01 > than the p value, 0.0025, then we would reject the null hypothesis. Therefore, At a 1% level of significance, there is sufficient evidence to conclude that the cost of attendance has increased.

180 square centimeters

step-by-step explanation: