The heights of a random sample of 50 Hereford calves at birth showed a mean of 71 cm and a standard deviation of 6.7 cm. Construct a 99% confidence interval for the mean height of all Hereford calves at birth. Give the lower limit.

Answer 1

To construct a 99% confidence interval for the mean height of all Hereford calves at birth, we will use the formula:

Confidence interval = mean ± (critical value * standard error)

The critical value can be found using the t-distribution table or a statistical calculator. For a 99% confidence interval with a sample size of 50 - 1 = 49 degrees of freedom, the critical value is approximately 2.68.

The standard error is calculated by dividing the standard deviation of the sample by the square root of the sample size:

Standard error = 6.7 cm / √50 ≈ 0.946 cm

Now we can calculate the confidence interval:

Confidence interval = 71 cm ± (2.68 * 0.946 cm)

Confidence interval = 71 cm ± 2.54 cm

To find the lower limit of the confidence interval, subtract the margin of error from the mean:

Lower limit = 71 cm - 2.54 cm

Lower limit ≈ 68.46 cm

Therefore, the lower limit of the 99% confidence interval for the mean height of all Hereford calves at birth is approximately 68.46 cm.