Question

In a study of larval development in the tufted apple budmoth (Platynota idaeusalis), an entomologist measured the head widths of 50 larvae. All 50 larvae had been reared under identical conditions and had moulted six times. The mean head width was \(1.20 \mathrm{~mm}\) and the standard deviation was \(0.14 \mathrm{~mm}\). Construct a \(90 \%\) confidence interval for the population mean. \(^{17}\)

Short answer

The 90% confidence interval for the population mean is (mean head width - t* x SE) to (mean head width + t* x SE), where mean head width is 1.20 mm, t* is the t-critical value for df=49 and 90% confidence level, and SE is the standard error calculated from the sample standard deviation and size.

Step-by-step solution

Determine the appropriate statistical test

Since we are constructing a confidence interval for the population mean based on a sample, we need to use a t-distribution if the population standard deviation is not known. Here, we will use the sample standard deviation and thus apply the t-distribution, as the sample size is relatively small (n=50) and the population standard deviation is not given.

Find the degrees of freedom

The degrees of freedom (df) for a t-distribution are found by subtracting 1 from the sample size (n). In this case, the degrees of freedom is 50 - 1 = 49.

Determine the t-critical value

For a 90% confidence level and 49 degrees of freedom, refer to the t-distribution table or use a calculator to find the t-critical value, which we will denote as t*. This is the value that the sample mean needs to exceed to be in the top 5% of a t-distribution (since 5% is left in the two tails combined for a 90% confidence interval).

Calculate the standard error

The standard error (SE) is the standard deviation divided by the square root of the sample size: SE = s / √n. Here, the standard deviation s is 0.14 mm and n=50.

Construct the confidence interval

The confidence interval is the range from (sample mean - t* x SE) to (sample mean + t* x SE). Plug in the mean head width, the t-critical value, and the standard error to calculate the interval.

Key concepts

Understanding the T-Distribution

The t-distribution, also known as the Student's t-distribution, is a probability distribution that is symmetrical and bell-shaped, much like the standard normal distribution. However, it has heavier tails, meaning it tends to produce values that fall further away from the mean, which is important when dealing with small sample sizes.

This comes into play especially when we do not know the true population standard deviation and instead have to rely on sample data. The t-distribution accounts for the additional uncertainty that comes with estimating the population standard deviation by the sample standard deviation. As the sample size increases, the t-distribution approaches the normal distribution. The t-distribution is critical in the construction of confidence intervals and hypothesis tests for population means when the sample size is small and the population standard deviation is unknown.

Degrees of Freedom Explained

Degrees of freedom, often abbreviated as df, are a measure of the amount of independent information available to estimate a parameter. In the context of the t-distribution, the degrees of freedom are equivalent to the number of independent observations minus the number of parameters being estimated.

For example, when estimating a population mean from a sample, the degrees of freedom are calculated as the sample size (n) minus one (df = n - 1). This subtraction accounts for the fact that we're using the sample mean to estimate one population parameter. The number of degrees of freedom affects the shape of the t-distribution; the larger the degrees of freedom, the closer the distribution will be to the normal distribution. In the scenario of the tufted apple budmoth larvae, with 50 larvae measured, we have 49 degrees of freedom.

Standard Error and its Significance

Standard error (SE) measures the precision of a sample mean as an estimator of the population mean. It quantifies how much the sample means you would expect to vary from the actual population mean if you were to take multiple samples from the population.

The formula for calculating the standard error is SE = s / √n, where 's' represents the sample standard deviation and 'n' is the sample size. A smaller standard, error then, indicates a more precise estimate of the population mean. It's crucial for determining the margin of error in a confidence interval; a narrower interval suggests a more accurate estimation. Given a standard deviation of 0.14 mm for the head widths of the larvae and a sample size of 50, one would compute the standard error to further facilitate the confidence interval calculation.

Population Mean Estimation

Estimating the population mean involves using sample data to infer the mean of the entire population from which the sample was taken. The point estimate for the population mean is the sample mean, but due to the uncertainty inherent in sampling, we also provide a range of values in which we are fairly confident the true population mean lies; this range is known as a confidence interval.

A confidence interval spans from the lower to the upper bound and is calculated using the formula (sample mean ± t* × SE), where 't*' is the t-critical value from the t-distribution for the specified confidence level and degrees of freedom. This interval estimates where the true population mean is likely to be, with a certain degree of confidence. In our exercise related to the tufted apple budmoth, we seek to construct a 90% confidence interval, which means we can say that, based on our sample, there is a 90% chance that the population mean head width falls within our calculated range.