Question

As part of a study of the development of the thymus gland, researchers weighed the glands of five chick embryos after 14 days of incubation. The thymus weights (mg) were as follows \(^{12}\) $$ \begin{array}{lllll} 29.6 & 21.5 & 28.0 & 34.6 & 44.9 \end{array} $$ For these data, the mean is 31.7 and the standard deviation is 8.7 (a) Calculate the standard error of the mean. (b) Construct a \(90 \%\) confidence interval for the population mean.

Short answer

The standard error of the mean is approximately 3.89 mg. The 90% confidence interval for the population mean is approximately (27.06, 36.34) mg.

Step-by-step solution

Calculate the Standard Error of the Mean

The standard error of the mean (SEM) is given by dividing the standard deviation (SD) by the square root of the sample size (n). SEM = \( \frac{SD}{\sqrt{n}} \).

Execute the Calculation for SEM

Given the standard deviation SD is 8.7 and the number of observations n is 5, the standard error of the mean (SEM) is calculated as follows: \( SEM = \frac{8.7}{\sqrt{5}} \approx \frac{8.7}{2.236} \approx 3.89 mg. \)

Determine the z-score for the 90% Confidence Interval

Using standard statistical tables, or a Z-distribution calculator, find the z-value that corresponds to a 90% confidence level. This value is approximately 1.645 for a one-sided test (since the distribution is symmetrical, it applies to both tails).

Calculate the Margin of Error for the Confidence Interval

The margin of error (E) is given by multiplying the standard error by the z-score. E = SEM * z-score. Substitute the values calculated in previous steps to obtain the margin of error.

Construct the 90% Confidence Interval for the Population Mean

The confidence interval is calculated as follows: Lower Limit = Mean - E; Upper Limit = Mean + E. Therefore, using the mean of 31.7 and the margin of error calculated, determine the interval.

Key concepts

Standard Error of the Mean

Understanding the standard error of the mean (SEM) is crucial when working with sample data to infer information about a population. The SEM is a measure of how much uncertainty there is in the estimate of the true mean. It's calculated by dividing the standard deviation (SD) by the square root of the sample size (n). This gives an idea of how spread out the sample means are likely to be if we were to collect multiple samples from the same population.

For example, with a standard deviation of 8.7 mg and a sample size of 5 chick embryos, the SEM is calculated as follows:
\( SEM = \frac{8.7}{\sqrt{5}} \) which results in approximately 3.89 mg. This relatively small sample size will likely lead to a larger SEM, indicating a higher uncertainty in estimating the true mean weight of the thymus glands.

Z-score

The z-score plays a pivotal role in the construction of confidence intervals for population parameters. It represents the number of standard deviations a data point is from the mean. When you construct a confidence interval, the z-score helps you to locate where a percentage of the most plausible values for the population mean lie on the normal distribution.

In the context of confidence intervals, the z-score corresponds to the desired level of confidence. For a 90% confidence interval, a z-score of approximately 1.645 is used. This means that we are looking at the value on the normal distribution such that 90% of the distribution is to the left of it, accounting for the fact that the normal distribution is symmetrical.

Sample Size

The sample size, often denoted as 'n', is the total number of observations in the sample. It is a key factor in statistics as it affects the precision of estimates such as the mean and the standard deviation of a population. Intuitively, larger samples tend to provide more accurate and reliable estimates of the population parameters because they are likely to reflect the population's variability better.

In the exercise, with a sample size of 5, the calculated SEM is higher than it would be with a larger sample size. This affects the width of the confidence interval, with smaller samples resulting in wider intervals and thus less precision in estimating the population mean.

Population Mean

The population mean is the average of values for a certain characteristic across the entire population. In practice, since it is often impossible to measure every individual in a population, researchers use sample means to estimate the population mean.

In our exercise, the mean of the thymus weights of 31.7 mg is a point estimate of the population mean. It is important to note that this is an estimate derived from the sample, and the true population mean might be different. A confidence interval will give a range in which we are fairly sure the population mean lies.

Margin of Error

The margin of error encompasses the extent of uncertainty in our estimate of the population mean. It provides a range around the sample mean within which the true population mean is likely to be found, with a certain level of confidence. The margin of error is calculated by multiplying the standard error of the mean by the z-score associated with our confidence level.

In our case, the margin of error is calculated as \( E = SEM \times z-score \), which with a SEM of approximately 3.89 mg and a z-score of 1.645 for a 90% confidence interval, yields a margin of error. This margin is then used to create an interval around the sample mean, suggesting where the true population mean might be located with 90% confidence.