A horticulturist wishes to estimate the mean growth of seedlings in a large timber plot last year. A random sample of n = 100 seedlings is selected and the one-year growth for each is measured. The sample yields = 5.62 cm. The known standard deviation for this popualtiopn is 2.50 cm. The 95 percent confidence interval for the mean growth is:
Question options:
(3.12, 8.12)
(4.98, 6.26)
(5.13, 6.11)
(5.37, 5.87)
(5.57, 5.67)
Uuh
(5.37, 5.87)
Hmm, let me calculate it for you. Just give me a moment.
Using the formula for the confidence interval:
Confidence interval = sample mean ± (critical value * standard error)
The critical value for a 95% confidence interval when the sample size is 100 is approximately 1.96.
The standard error can be calculated as the population standard deviation divided by the square root of the sample size: 2.50 / √100 = 0.25.
Plugging in the values:
Confidence interval = 5.62 ± (1.96 * 0.25)
Calculating the upper and lower limits:
Lower limit = 5.62 - (1.96 * 0.25) = 5.12
Upper limit = 5.62 + (1.96 * 0.25) = 6.12
So, the 95% confidence interval for the mean growth is (5.12, 6.12).
Therefore, the correct answer is option (5.13, 6.11).
To find the 95% confidence interval for the mean growth, we can use the formula:
Confidence Interval = mean ± (Z * (standard deviation / √n))
For a 95% confidence level, the Z-value is approximately 1.96.
Given:
Sample mean (x̄) = 5.62 cm
Known standard deviation (σ) = 2.50 cm
Sample size (n) = 100 seedlings
Plugging in the values into the formula, we get:
Confidence Interval = 5.62 ± (1.96 * (2.50 / √100))
Calculating the value inside the parentheses:
Confidence Interval = 5.62 ± (1.96 * 0.25)
Confidence Interval = 5.62 ± 0.49
Calculating the upper and lower limits of the confidence interval:
Upper limit = 5.62 + 0.49 = 6.11
Lower limit = 5.62 - 0.49 = 5.13
Therefore, the 95% confidence interval for the mean growth is (5.13, 6.11).
Formula:
CI95 = mean ± 1.96(sd/√n)
Your data:
mean = 5.62
sd = 2.50
n = 100
Substitute the values into the formula and calculate the interval. You should then be able to select the correct answer.