Question 1179489: The height in centimeters of a certain plant is normally distributed. A random sample
of the plant is measured and the results are as follows:
14.6;12.5;15.3; 16.1; 14.4; 12.9; 13.7; 14.9
a. Find a point estimate for the mean height of this plant.
b. Construct a 90% confidence interval for the true mean height of this particular
plant.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **a) Point Estimate for the Mean Height**
The sample mean (x̄) is a good point estimate for the population mean (μ).
To calculate the sample mean, add up all the heights and divide by the number of plants in the sample:
x̄ = (14.6 + 12.5 + 15.3 + 16.1 + 14.4 + 12.9 + 13.7 + 14.9) / 8
x̄ = 114.4 / 8
**x̄ = 14.3 cm**
**b) 90% Confidence Interval for the True Mean Height**
Since the population standard deviation is unknown, we'll use a t-distribution to construct the confidence interval.
**1. Calculate the Sample Standard Deviation (s)**
s = √[ Σ(xi - x̄)² / (n - 1) ]
where:
* xi = each individual height
* x̄ = sample mean
* n = sample size
s ≈ 1.283 cm (You can use a calculator or software to compute this)
**2. Determine the Degrees of Freedom (df)**
df = n - 1 = 8 - 1 = 7
**3. Find the t-value**
For a 90% confidence interval and 7 degrees of freedom, we need the t-value that leaves 5% in each tail (α/2 = 0.10/2 = 0.05).
Using a t-table or calculator, the t-value is approximately 1.895.
**4. Calculate the Margin of Error (E)**
E = t * (s / √n) = 1.895 * (1.283 / √8) ≈ 0.861 cm
**5. Construct the Confidence Interval**
Confidence Interval = x̄ ± E = 14.3 ± 0.861
Lower Bound = 14.3 - 0.861 = 13.439 cm
Upper Bound = 14.3 + 0.861 = 15.161 cm
**Therefore, the 90% confidence interval for the true mean height of the plant is approximately (13.44 cm, 15.16 cm).**
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