You measure 39 dogs' weights, and find they have a mean weight of 76 ounces. Assume the population standard deviation is 12.1 ounces. Based on this, construct a 90% confidence interval for the true population mean dog weight.
To construct a 90% confidence interval for the true population mean dog weight, we can use the formula:
Confidence interval = mean ± (Z * (standard deviation / √n))
We were given the following information:
- Mean weight of the 39 dogs: 76 ounces
- Population standard deviation: 12.1 ounces
Since we are constructing a 90% confidence interval, the level of significance is 1 - 0.90 = 0.10. We need to find the critical value (Z) for a one-tailed test at a significance level of 0.10.
Using a standard normal distribution table or calculator, we find that the critical value for a 90% confidence interval is approximately 1.645.
Substituting the values into the formula, we have:
Confidence interval = 76 ± (1.645 * (12.1 / √39))
Calculating the square root of 39:
√39 ≈ 6.2449979984
Substituting this value into the formula:
Confidence interval = 76 ± (1.645 * (12.1 / 6.2449979984))
Calculating 12.1 / 6.2449979984:
12.1 / 6.2449979984 ≈ 1.9366286663
Substituting this value into the formula:
Confidence interval = 76 ± (1.645 * 1.9366286663)
Calculating 1.645 * 1.9366286663:
1.645 * 1.9366286663 ≈ 3.1831897576
Thus, the confidence interval for the true population mean dog weight is:
Confidence interval = 76 ± 3.183
Therefore, the 90% confidence interval is approximately (72.817, 79.183).