You measure 47 dogs' weights, and find they have a mean weight of 75 ounces. Assume the population standard deviation is 12.4 ounces. Based on this, construct a 90% confidence interval for the true population mean dog weight.
Round your answers to two decimal places.
To construct a confidence interval for the true population mean dog weight, we can use the formula:
Confidence interval = sample mean ± (critical value) * (standard deviation of the sample)
First, we need to find the critical value for a 90% confidence interval. Since we have a large sample size (n > 30), we can use the Z-distribution. The critical value for a 90% confidence interval is Z = 1.645.
Next, we need to calculate the standard deviation of the sample, which is also known as the standard error. To find the standard error, we divide the population standard deviation by the square root of the sample size:
Standard error (SE) = standard deviation / √(sample size)
SE = 12.4 / √47
SE ≈ 1.80
Now we can construct the confidence interval:
Confidence interval = sample mean ± (critical value) * (standard error)
Confidence interval = 75 ± (1.645 * 1.80)
Confidence interval ≈ 75 ± 2.96
Rounded to two decimal places, the confidence interval for the true population mean dog weight is:
Confidence interval ≈ (72.04, 77.96)