Question

1. In a study on infants, one of the characteristics measured was head circumference. The mean head circumference of 14 infants was 34.6 cm.

   1. Assuming that head circumferences for infants are normally distributed with standard deviation 2.1 cm, determine a 95% confidence interval for the mean head circumference of all infants.

      Step 1: Determine ??/2. Step 2: Determine the CI.

      Step 3: Interpret the CI.

   2. Obtain the margin of error, ?, for the CI you found in part (a).

   3. Determine the sample size required to have a margin of error of 0.7 cm with a 95% confidence level.

2. The following data represent the concentration of organic carbon (mg/L collected from organic soil. Construct a 99% confidence interval for the mean concentration of dissolved organic carbon collected from organic soil. Note that ? = 17.63mg/L and ? = 7.66mg/L

5.30 29.80 27.10 16.51   15.72

8.81 16.87 20.46 14.90 33.67

30.91 14.86 17.50 15.35   9.72

19.80 14.86 8.09 14.00 18.30

Step 1: Compute ?/2 for ? = (1 − confidence level) and determine the sample mean and sample standard deviation.

Step 2: Use Table IV to find ??/2 with ?? = ? − 1.

Step 3: Determine the CI.

Step 4: Interpret the CI.

Answer 1

1)

1)

sample mean, xbar = 34.6
sample standard deviation, σ = 2.1
sample size, n = 14

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

ME = zc * σ/sqrt(n)
ME = 1.96 * 2.1/sqrt(14)
ME = 1.1

2)

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (34.6 - 1.96 * 2.1/sqrt(14) , 34.6 + 1.96 * 2.1/sqrt(14))
CI = (33.5 , 35.7)

3)

we are 95% confident taht the mean head circumference of all infants is between (33.5 , 35.7)

The following information is provided,
Significance Level, α = 0.05, Margin or Error, E = 0.7, σ = 2.1

The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.96 * 2.1/0.7)^2
n = 34.57

Therefore, the sample size needed to satisfy the condition n >= 34.57 and it must be an integer number, we conclude that the minimum required sample size is n = 35
Ans : Sample size, n = 35

2)

1(
sample mean, xbar = 17.63
sample standard deviation, s = 7.66

2)

sample size, n = 20
degrees of freedom, df = n - 1 = 19

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, tc = t(α/2, df) = 2.861

3)

ME = tc * s/sqrt(n)
ME = 2.861 * 7.66/sqrt(20)
ME = 4.9004

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (17.63 - 2.861 * 7.66/sqrt(20) , 17.63 + 2.861 * 7.66/sqrt(20))
CI = (12.7296 , 22.5304)

4)

we are 99% confident that the mean concentration of dissolved organic carbon collected from organic soil is between (12.7296 , 22.5304)