Dairy cows at large commercial farms often receive...

Dairy cows at large commercial farms often receive injections of bST (Bovine Somatotropin), a hormone used to spur milk production. Bauman et al. (Journal of Dairy Science, 1989) reported that 12 cows given bST produced an average of $28.0 \mathrm{~kg} / \mathrm{d}$ of milk. Assume that the standard deviation of milk production is $2.25 \mathrm{~kg} / \mathrm{d}$ (a) Find a $99 \%$ confidence interval for the true mean milk production. (b) If the farms want the confidence interval to be no wider than $\pm 1.25 \mathrm{~kg} / \mathrm{d},$ what level of confidence would they need to use?

Dairy cows at large commercial farms often receive injections of bST (Bovine Somatotropin), a hormone used to spur milk production. Bauman et al. (Journal of Dairy Science,
1989) reported that 12 cows given bST produced an average of $28.0 \mathrm{~kg} / \mathrm{d}$ of milk. Assume that the standard deviation of milk production is $2.25 \mathrm{~kg} / \mathrm{d}$
(a) Find a $99 \%$ confidence interval for the true mean milk production.
(b) If the farms want the confidence interval to be no wider than $\pm 1.25 \mathrm{~kg} / \mathrm{d},$ what level of confidence would they need to use?

Key Concepts

Standard Error

The standard error measures the variability or dispersion of the sample mean from the true population mean. It is calculated as the population standard deviation (or an estimate thereof) divided by the square root of the sample size. This concept is critical when constructing confidence intervals because it quantifies the precision of the estimated mean.

t-Distribution

The t-distribution is used in situations where the sample size is small and/or the population standard deviation is unknown. It accounts for the additional uncertainty inherent in using a sample to estimate the population parameter. The shape of the t-distribution is similar to the normal distribution but has heavier tails, which means it provides a wider interval for smaller sample sizes.

Confidence Level

The confidence level represents the probability that the confidence interval calculated from a random sample will contain the true population parameter. Common confidence levels include 90%, 95%, and 99%. A higher confidence level results in a wider interval, reflecting increased certainty that the true parameter lies within the interval, while a lower confidence level results in a narrower interval.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. This concept is fundamental in inferential statistics, as it provides an estimate of the parameter and quantifies the uncertainty around the estimate by specifying the degree of confidence (e.g., 99%) that the interval includes the true parameter.

Margin of Error

The margin of error defines the extent of the interval on either side of the sample statistic, indicating the maximum expected difference between the sample estimate and the true population parameter. It is calculated using the critical value (from the t- or z-distribution) and the standard error of the sample, and it directly influences the width of the confidence interval.

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