Question

Is the nutrition information on commercially produced food accurate? In one study, researchers sampled 13 packages of a certain frozen reduced-calorie chicken entrée with a reported calorie content of 252 calories per package. The mean calorie count of the sampled entrées was 306 with a sample standard deviation of 51 calories. \(^{59}\) (a) Compute a \(90 \%\) confidence interval for the population mean calorie content of the frozen entrée. (b) Compute a \(99 \%\) confidence interval for the population mean calorie content of the frozen entrée. (c) Based on the two intervals computed in parts (a) and (b), what do you think about the reported calorie content for this entrée?

Short answer

After calculating, we find that the 90% CI does not contain 252 calories, nor does the 99% CI, suggesting the reported calorie content may be understated.

Step-by-step solution

Determine the confidence level and degrees of freedom

To compute the confidence intervals, first identify the confidence levels required for each part. For part (a) it's 90% confidence level, and for part (b) it's 99%. The degrees of freedom (df) for the t-distribution is the sample size minus one, which is 13 - 1 = 12.

Find the appropriate t-critical values

Using a t-distribution table or calculator, find the t-critical values for a 90% and 99% confidence level with 12 degrees of freedom. For a two-tailed test, divide the alpha level by 2. The t-critical values correspond to the tails of the distribution beyond which we can expect to find a combined 10% or 1% of the distribution for the 90% and 99% confidence intervals, respectively.

Calculate the Standard Error

The standard error (SE) of the mean is calculated using the sample standard deviation (s) and the sample size (n). SE = s / sqrt(n). For our data, SE = 51 / sqrt(13).

Compute the Margin of Error for 90% CI

The margin of error (ME) is the product of the t-critical value for 90% confidence and the standard error. ME = t-critical(90%) * SE.

Compute the 90% Confidence Interval

To compute the confidence interval, add and subtract the margin of error from the sample mean. 90% CI = [mean - ME, mean + ME].

Compute the Margin of Error for 99% CI

The margin of error for the 99% confidence interval is calculated similarly to the 90% confidence interval, using the t-critical value for 99%. ME = t-critical(99%) * SE.

Compute the 99% Confidence Interval

Just like with the 90% confidence interval, add and subtract the margin of error for the 99% CI from the sample mean. 99% CI = [mean - ME, mean + ME].

Interpret the Confidence Intervals

Compare the reported calorie content of 252 calories per package with the computed confidence intervals. If the reported value does not fall within these intervals, this could suggest that the reported calorie content might not be accurate.

Key concepts

Confidence Level

When we calculate a confidence interval, we choose a confidence level that indicates how certain we are that the true parameter lies within the interval we calculate. A common confidence level used is 90%, but it could be higher like 95% or 99% depending on how confident the researcher wants to be about their results.

In the context of the provided exercise, two confidence levels are used - 90% for part (a) and 99% for part (b). This implies that for the 90% confidence interval, there is a 10% chance that the true mean calorie content does not fall within our calculated interval. Similarly, for the 99% confidence interval, there is only a 1% chance that the true mean lies outside of this interval. In simpler terms, a higher confidence level means the interval will be wider to be more certain that it contains the true mean.

Degrees of Freedom

The concept of degrees of freedom (df) is paramount when dealing with t-distributions in confidence interval calculations. Degrees of freedom refer to the number of values that are free to vary in a dataset while calculating a statistic.

In the step by step solution, we find that the degrees of freedom for our sample is the sample size minus one (\( df = n - 1 \)), yielding us 12 degrees of freedom for our set of 13 chicken entrées. The degrees of freedom are critical when it comes to finding the t-critical values from t-distribution tables, as they determine the shape of the t-distribution curve we will be using to find our margin of error.

T-Distribution

The t-distribution is a type of probability distribution that is symmetrical and bell-shaped, similar to the standard normal distribution, but with thicker tails. These thicker tails are due to higher variability expected with small sample sizes.

For calculating our confidence intervals in the exercise, we need t-critical values, which are derived from the t-distribution taking into account the degrees of freedom and the desired confidence level. The further out in the tails you go (higher confidence levels), the larger the t-critical value will be. When sample sizes are large, the t-distribution closely resembles the standard normal distribution.

Standard Error

Standard error (SE) is a measure that indicates how far the sample mean (the average from your sample data) is likely to be from the true population mean.

It is derived from the sample standard deviation (\( s \)) and the square root of the sample size (\( n \[\begin{equation}SE = \frac{s}{\sqrt{n}}\end{equation}\]\[\begin{equation}SE = \frac{51}{\sqrt{13}}\end{equation}\]\)). In our exercise, standard error plays a vital role as it's used to calculate the margin of error, which determines the width of our confidence interval. In essence, a smaller standard error suggests that there is less variability in the sample means, which typically results in a narrower confidence interval.

Margin of Error

Margin of error (ME) helps us define the range around the sample mean within which we expect the true population mean to lie. This range accounts for potential sampling errors and is dependent on both the t-critical value (which takes into account the confidence level and degrees of freedom) and the standard error.

We calculate it by multiplying the standard error by the t-critical value for the desired confidence level (\( ME = t\text{-critical} * SE \[\begin{equation}ME_{90\%} = t_{90\%} * SE\end{equation}\]\[\begin{equation}ME_{99\%} = t_{99\%} * SE\end{equation}\]\)). In the exercise, the margin of error is what we add and subtract from the sample mean to get the lower and upper bounds of the confidence interval. The larger the margin of error, the wider the interval, and this wideness indicates more uncertainty about the exact value of the population mean.