I want to estimate the average time it takes a tomato plant to bear fruit. I randomly sample 7 tomato plants and get an average of 91 days. Assume 𝜎=12 days. Find a 99% confidence interval for the true average time to grow a tomato.Use 1 decimal place.
Question
I want to estimate the average time it takes a tomato plant to bear fruit. I randomly sample 7 tomato plants and get an average of 91 days. Assume 𝜎=12 days. Find a 99% confidence interval for the true average time to grow a tomato.Use 1 decimal place.
Solution
To calculate a 99% confidence interval, you will need to use the formula for a confidence interval which is:
CI = x̄ ± Z * (σ/√n)
where:
- x̄ is the sample mean
- Z is the Z-score (which corresponds to the desired confidence level)
- σ is the standard deviation
- n is the sample size
Given in the problem, we have:
- x̄ = 91 days
- σ = 12 days
- n = 7
The Z-score for a 99% confidence interval is approximately 2.576 (you can find this value in a standard Z-table or using a calculator that can calculate Z-scores).
Substituting these values into the formula, we get:
CI = 91 ± 2.576 * (12/√7)
To simplify this, first calculate the standard error (σ/√n):
SE = 12/√7 ≈ 4.53
Then multiply the Z-score by the standard error:
Margin of Error = Z*SE = 2.576 * 4.53 ≈ 11.7
So, the 99% confidence interval is:
CI = 91 ± 11.7
This gives us a confidence interval of (79.3, 102.7).
So, we can be 99% confident that the true average time it takes for a tomato plant to bear fruit is between 79.3 and 102.7 days.
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