Question

Many consumers pay careful attention to stated nutritional contents on packaged foods when making purchases. It is therefore important that the information on packages be accurate. A random sample of \(n=12\) frozen dinners of a certain type was selected from production during a particular period, and the calorie content of each one was determined. (This determination entails destroying the product, so a census would certainly not be desirable!) Here are the resulting observations, along with a boxplot and normal probability plot: \(\begin{array}{llllllll}255 & 244 & 239 & 242 & 265 & 245 & 259 & 248\end{array}\) \(\begin{array}{llll}225 & 226 & 251 & 233\end{array}\) a. Is it reasonable to test hypotheses about mean calorie content \(\mu\) by using a \(t\) test? Explain why or why not. b. The stated calorie content is \(240 .\) Does the boxplot suggest that true average content differs from the stated value? Explain your reasoning. c. Carry out a formal test of the hypotheses suggested in Part (b).

Short answer

Yes, it's reasonable to apply t-test after verifying normality of the distribution. Interpretation of boxplot for mean comparison is subjective and depends on IQR and mean placement. The formal t-test involves stating hypotheses, calculating t-statistic and comparing it with critical value to take decision about stated calorie content validity.

Step-by-step solution

Consideration for using t-test

A t-test can be used for hypothesis testing when we have a small sample size and the population standard deviation is unknown. These conditions fit our given situation as we have sample size (n) of 12 and we don't know the population standard deviation. Moreover, the t-test also requires the dataset to be approximately normally distributed, which is checked by creating a normal probability plot.

Analyzing Boxplot for mean comparison

To decide if the boxplot indicates that the true average differs from the stated value, visually observe the placement of the mean (or median in some cases) in the boxplot. If the stated value (240) is out of the interquartile range (the box), or if it is distant from the mean, it suggests a possible difference between the true average and stated value.

Performing t-test

To perform the t-test, begin by stating the null and alternative hypotheses.\nThe null hypothesis (Ho): \( \mu = 240 \) kcal (the true mean calorie content is equal to the stated value).\nThe alternative hypothesis (Ha): \( \mu \neq 240 \) kcal (the true mean calorie content is not equal to the stated value).\n\nNow calculate the sample mean and standard deviation. Use these values, along with the sample size (n=12), to calculate the t-statistic by the formula: \( t = \frac{\bar{x}-\mu}{s/ \sqrt{n}} \)\n\nCompare the calculated t-statistic with the t-critical value from the t-distribution table for \( \alpha=0.05 \) (or any other level of significance chosen) and df = n-1. If the test statistic t is greater or lesser than the critical value (depends on one-tailed or two-tailed test), reject Ho. If not, fail to reject Ho. This will validate or invalidate the stated calorie content on the packages.

Key concepts

Normal Probability Plot

The normal probability plot is a graphical tool used to assess if a dataset is approximately normally distributed. It’s a key assumption for many statistical tests, including the t-test. When you create this plot, you’re essentially plotting the observed data against a theoretical normal distribution. The resulting plot shows how closely your data fits a normal distribution.

How to read it? If the points fall roughly along a straight line, your data is likely normally distributed. If they stray far from this line, normality is questionable. In our frozen dinner scenario, the normal probability plot would help determine if the calorie data can be appropriately analyzed using a t-test, which requires that the data be roughly normal.

Boxplot

A boxplot is a simple yet powerful way to visualize the distribution of a dataset. It displays the dataset’s central tendency and variability and shows possible outliers.

In a boxplot, the box represents the interquartile range (IQR), which contains the middle 50% of the data. The line in the box marks the median, while the "whiskers" indicate variability outside the upper and lower quartiles. Outliers may appear as individual points beyond the whiskers.

In analyzing our frozen dinners, if the calorie count of 240 is outside the box or distant from the median line, it suggests a possible difference in the stated average calorie content. This could hint that the actual average calorie count of the sampled dinners is different from what’s advertised.

Hypothesis Testing

Hypothesis testing is a method to decide if a specific statement about a population parameter is supported by sample data. Consider two hypotheses: the null hypothesis \[ H_0: \mu = 240 \](meaning the true mean is the same as the stated value) and the alternative hypothesis\[ H_a: \mu eq 240 \](meaning the true mean differs).

To test these hypotheses, we use a t-test. This involves calculating a test statistic based on the sample mean, standard deviation, and sample size. Comparing this value with a critical value from a t-distribution table helps us understand if the data significantly supports the null hypothesis or if we have enough evidence to consider the alternative.

Sample Mean

The sample mean is the average of all data points in a sample. It’s a crucial measure in statistics, representing an estimate of the population mean. Calculating the sample mean involves adding all the data points together and dividing by the number of points.

Let’s say the calorie counts of our dinners are summed and divided by 12. The result is the sample mean. This value is then compared to the stated calorie content of 240. Differences between the sample mean and the stated value can imply discrepancies in labeling. The sample mean is the foundation for further analyses, like deriving the test statistic in hypothesis testing.