Question 1179489
**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).**