Question

For the 28 lamb birthweights of Example \(6.2 .2,\) the mean is \(5.1679 \mathrm{~kg},\) the \(\mathrm{SD}\) is \(0.6544 \mathrm{~kg},\) and the \(\mathrm{SE}\) is \(0.1237 \mathrm{~kg}\) (a) Construct a \(95 \%\) confidence interval for the population mean. (b) Construct a \(99 \%\) confidence interval for the population mean. (c) Interpret the confidence interval you found in part (a). That is, explain what the numbers in the interval mean. (Hint: See Examples 6.3 .4 and \(6.3 .5 .)\) (d) Often researchers will summarize their data in reports and articles by writing \(\bar{y} \pm \mathrm{SD}(5.17 \pm 0.65)\) or \(\bar{y} \pm \mathrm{SE}(5.17 \pm 0.12) .\) If the researcher of this study is planning to compare the mean birthweight of these Rambouillet lambs to another breed, Booroolas, which style of presentation should she use?

Short answer

95% CI: (4.9233, 5.4125) kg, 99% CI: (4.8524, 5.4834) kg. Interpretation: There is 95% confidence that the population mean birthweight of the lambs is between 4.9233 and 5.4125 kg. It's preferable to use \(\bar{y} \pm SE\) for comparing means.

Step-by-step solution

Identify the Appropriate Z-values

For the 95% confidence interval, the Z-value corresponding to the 95% confidence level is found from a Z-table or standard normal distribution table, which is approximately 1.96. For the 99% confidence interval, the Z-value is approximately 2.576.

Calculate the 95% Confidence Interval

Use the formula for the confidence interval: \(\text{CI} = \bar{x} \pm Z \cdot SE\). We calculate the 95% CI using the mean (\(\bar{x}\)), the Z-value for 95% (1.96), and the standard error (SE). So, the 95% CI is \((5.1679 \pm 1.96 \cdot 0.1237)\) kg.

Calculate the 99% Confidence Interval

Similarly, we calculate the 99% CI using the Z-value for 99% (2.576) with the same mean and SE. The 99% CI is \((5.1679 \pm 2.576 \cdot 0.1237)\) kg.

Interpret the 95% Confidence Interval

The 95% confidence interval means that we are 95% confident that the population mean of lamb birthweights falls within the calculated interval. It does not mean that 95% of the individual lamb birthweights will fall within this interval.

Recommend a Style of Presentation

Since the researcher is comparing the mean birthweight with another breed, it's more appropriate to use the mean plus or minus the standard error (\(5.17 \pm 0.12\)). This is because the SE reflects how much the sample mean is expected to vary from the true population mean and is suitable for comparing means between groups.

Key concepts

Standard Error

Understanding the standard error (SE) is vital when delving into the realms of statistics, especially when constructing confidence intervals for population parameters. In simple terms, the standard error measures the accuracy with which a sample represents a population. It is derived from the standard deviation (SD) and the sample size (n), following the formula: SE = \( \frac{SD}{\sqrt{n}} \).

The smaller the standard error, the more representative the sample mean is likely to be of the population mean. When calculating confidence intervals, the SE is used in conjunction with a Z-value to gauge the range within which the true population mean is expected to lie. It's essential to note that it reflects not the variation within the data itself, but the potential variation of the sample mean if the study were repeated multiple times.

Population Mean

The population mean is a key concept in statistics representing the average value in a complete set. While this number is often unknown in real-world scenarios, researchers use sample data to estimate it. Consider the case where a study of lamb birthweights reports a mean (\(\bar{x}\)) of 5.1679 kg. This figure suggests that the average birthweight of all lambs, assuming the sample is representative of the larger population of Rambouillet lambs, might be close to this value.

Since it's impractical to weigh every single lamb, we use the sample mean as the best estimation of the population mean. Through the construction of confidence intervals, researchers can express how confident they are that their estimation is accurate—and that's precisely where the population mean becomes a focal point.

Z-value

The Z-value plays a crucial role when constructing confidence intervals around a population mean. It is a statistic that represents the number of standard errors a point is from the mean of a standard normal distribution. In other words, it's a way of quantifying how 'unusual' a data point is within the context of a normal distribution.

To find appropriate Z-values, researchers refer to Z-tables that align with desired confidence levels like 95% or 99%. For instance, a Z-value of 1.96 corresponds to a confidence level of 95%, meaning that the true population mean is expected to lie within 1.96 SEs of the sample mean 95% of the time. As the confidence level increases to 99%, a higher Z-value of 2.576 is used, reflecting the wider interval needed for greater certainty.

Statistical Significance

Statistical significance is a determination of whether a result is likely due to something other than chance. In the context of confidence intervals and hypothesis testing, it helps to discern whether the observed differences or relationships are 'real' or could have occurred randomly. When an interval does not include a certain value (often zero), the result can be deemed statistically significant.

In our example, after calculating the confidence interval, we may compare it against a known value or another study to ascertain significance. If the interval does not overlap with this value or range, we may infer that the observed difference is statistically significant. This is a crucial step for researchers concluding their analysis and establishing whether their findings have both practical and statistical merit.