A random sample of 36 lemons is selected from an orchard and the volume of juice each contains is measured. The sample mean is 241 milliliters and the sample standard deviation is 12 milliliters.A 95% confidence interval for the true mean volume of juice in a lemon from this orchard is:Select one:a.from 237ml to 245ml.b.from 217ml to 265ml.c.from 239ml to 243ml.d.from 229ml to 253ml.
Question
A random sample of 36 lemons is selected from an orchard and the volume of juice each contains is measured. The sample mean is 241 milliliters and the sample standard deviation is 12 milliliters.A 95% confidence interval for the true mean volume of juice in a lemon from this orchard is:Select one:a.from 237ml to 245ml.b.from 217ml to 265ml.c.from 239ml to 243ml.d.from 229ml to 253ml.
Solution
A 95% confidence interval for the mean is calculated using the formula:
CI = x̄ ± (Z * (σ / √n))
where: CI = confidence interval x̄ = sample mean Z = z-score corresponding to the desired confidence level (for a 95% confidence level, Z = 1.96) σ = standard deviation n = sample size
In this case, the sample mean (x̄) is 241 milliliters, the standard deviation (σ) is 12 milliliters, and the sample size (n) is 36.
Substituting these values into the formula gives:
CI = 241 ± (1.96 * (12 / √36))
Calculating the square root of 36 gives 6.
So, CI = 241 ± (1.96 * (12 / 6))
This simplifies to:
CI = 241 ± (1.96 * 2)
So, CI = 241 ± 3.92
This gives a confidence interval from 237.08 milliliters to 244.92 milliliters.
Rounding to the nearest whole number, the 95% confidence interval for the true mean volume of juice in a lemon from this orchard is from 237 milliliters to 245 milliliters.
So, the correct answer is:
a. from 237ml to 245ml.
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