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.
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.
To construct a 99% confidence interval for the mean height of all Hereford calves at birth, we can use the formula:
Confidence Interval = mean ± (critical value) * (standard deviation / sqrt(sample size))
First, we need to find the critical value using the Z-table or a calculator. For a 99% confidence level, the critical value is 2.576.
Next, we can substitute the values into the formula:
Confidence Interval = 71 ± (2.576) * (6.7 / sqrt(50))
Calculating the square root of 50 gives us approximately 7.071.
Confidence Interval ≈ 71 ± (2.576) * (6.7 / 7.071)
Simplifying further:
Confidence Interval ≈ 71 ± (2.576) * (0.948)
Confidence Interval ≈ 71 ± (2.443)
Now, we can find the lower limit of the confidence interval by subtracting the value obtained from 71:
Lower Limit ≈ 71 - 2.443
Lower Limit ≈ 68.557
Therefore, the lower limit of the 99% confidence interval for the mean height of all Hereford calves at birth is approximately 68.557 cm.