Use the confidence level and sample data to find a confidence interval for estimating the population μ. Round your answer to the same number of decimal places as the sample mean.

A random sample of 130 full-grown lobsters had a mean weight of 21 ounces and a standard deviation of 3.0 ounces. Construct a 98% confidence interval for the population mean μ.

A) 20 oz < μ < 22 oz
B) 21 oz < μ < 23 oz
C) 19 oz < μ < 21 oz
D) 20 oz < μ < 23 oz

1 answer

To construct a 98% confidence interval for the population mean (μ), we can use the formula for the confidence interval when the population standard deviation is unknown and the sample size is large (n > 30), which allows us to use the t-distribution.

  1. Identify the sample statistics:

    • Sample mean (\(\bar{x}\)) = 21 ounces
    • Sample standard deviation (s) = 3.0 ounces
    • Sample size (n) = 130
  2. Determine the critical value: For a 98% confidence level and a sample size of 130, we need to find the t-score that corresponds to \(\alpha/2\) (where \(\alpha = 0.02\)) for \(n - 1 = 129\) degrees of freedom. However, for large sample sizes, the z-score can be used as an approximation. The z-score for a 98% confidence level can be found using a z-table or normal distribution calculator.

    The z-score for 98% confidence is approximately 2.33.

  3. Calculate the standard error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{3.0}{\sqrt{130}} \approx \frac{3.0}{11.401} \approx 0.263 \]

  4. Calculate the margin of error (ME): \[ ME = z \times SE \approx 2.33 \times 0.263 \approx 0.614 \]

  5. Construct the confidence interval: \[ \text{Confidence interval} = \bar{x} \pm ME \] \[ \text{Confidence interval} = 21 \pm 0.614 \] This results in:

    • Lower limit: \(21 - 0.614 \approx 20.386\)
    • Upper limit: \(21 + 0.614 \approx 21.614\)
  6. Round and present the final confidence interval: Rounding to the same number of decimal places as the sample mean (which is 21 ounces), we get: \[ 20.4 \text{ oz} < μ < 21.6 \text{ oz} \]

Now, let's compare our result with the available options:

A) 20 oz < μ < 22 oz (This interval does include our result.) B) 21 oz < μ < 23 oz C) 19 oz < μ < 21 oz D) 20 oz < μ < 23 oz

The best option that contains our calculated interval (20.4 oz < μ < 21.6 oz) is:

A) 20 oz < μ < 22 oz

Thus, the answer is A.