To construct a confidence interval, we can use the formula:
Confidence Interval = sample mean ± (critical value) x (standard deviation/square root of sample size)
Given:
Sample mean (x̄) = 2.6 milligrams
Standard deviation (σ) = 0.9 milligrams
Sample size (n) = 8
First, we need to find the critical value for a 99% confidence level.
Since the sample size is small (n < 30), we use a t-distribution instead of a z-distribution.
Degrees of freedom (df) = n - 1
df = 8 - 1 = 7
Using a t-table or calculator, the critical value for a 99% confidence level with df = 7 is approximately 3.499.
Now, we can calculate the confidence interval:
Confidence Interval = 2.6 ± (3.499) x (0.9/√8)
To calculate the standard error (standard deviation/square root of sample size):
Standard Error = 0.9/√8 ≈ 0.318
Confidence Interval = 2.6 ± (3.499) x (0.318)
Calculating the values:
Lower limit = 2.6 - (3.499) x (0.318) ≈ 2.6 - 1.117 ≈ 1.483
Therefore, the lower limit of the 99% confidence interval is approximately 1.483 milligrams.
A random sample of 8 cigarettes of a certain brand has an average nicotine content of 2.6 milligram and standard deviation of 0.9 milligrams. Construct a 99% confidence interval for the true average nicotine content of this brand of cigarettes, assuming the distribution of nicotine content to be normal. Give the lower limit.
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