A random sample of 12 items is taken and is found to have a meanweight of 50 grams and a standard deviation of 9 gramsWhat is the mean weight of population1. with 95% confidence2. with 99% confidence
Question
A random sample of 12 items is taken and is found to have a meanweight of 50 grams and a standard deviation of 9 gramsWhat is the mean weight of population1. with 95% confidence2. with 99% confidence
Solution 1
To calculate the confidence interval for the mean weight of the population, we use the formula:
Confidence Interval = X ± Z * (s/√n)
where: X = sample mean Z = Z score (which varies depending on the confidence level) s = standard deviation n = sample size
- With 95% confidence: The Z score for a 95% confidence level is approximately 1.96.
Substituting the given values into the formula, we get:
Confidence Interval = 50 ± 1.96 * (9/√12)
Calculate the value inside the parenthesis:
= 50 ± 1.96 * (9/3.46) = 50 ± 1.96 * 2.6 = 50 ± 5.1
So, the mean weight of the population with 95% confidence is between 44.9 grams and 55.1 grams.
- With 99% confidence: The Z score for a 99% confidence level is approximately 2.58.
Substituting the given values into the formula, we get:
Confidence Interval = 50 ± 2.58 * (9/√12)
Calculate the value inside the parenthesis:
= 50 ± 2.58 * (9/3.46) = 50 ± 2.58 * 2.6 = 50 ± 6.7
So, the mean weight of the population with 99% confidence is between 43.3 grams and 56.7 grams.
Solution 2
To calculate the confidence interval for the mean weight of the population, we use the formula:
Confidence Interval = X ± Z * (s/√n)
where: X = sample mean Z = Z score (value from the standard normal distribution corresponding to desired confidence level) s = standard deviation n = sample size
- With 95% confidence: The Z score for 95% confidence is approximately 1.96.
Confidence Interval = 50 ± 1.96 * (9/√12) = 50 ± 1.96 * (9/3.46) = 50 ± 1.96 * 2.6 = 50 ± 5.1 So, the mean weight of the population with 95% confidence is between 44.9 grams and 55.1 grams.
- With 99% confidence: The Z score for 99% confidence is approximately 2.58.
Confidence Interval = 50 ± 2.58 * (9/√12) = 50 ± 2.58 * (9/3.46) = 50 ± 2.58 * 2.6 = 50 ± 6.7 So, the mean weight of the population with 99% confidence is between 43.3 grams and 56.7 grams.
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