For each of the following questions, decide whether or not the sample data can be used to construct a confidence interval.

How do I go about determining if the sample data can be used to construct a confidence interval?

Example:

To help consumers assess the risks they are taking, the Food and Drug Administration (FDA) publishes the amount of nicotine found in all commercial brands of cigarettes. A new cigarette has recently been marketed. The FDA tests on this cigarette gave a mean nicotine content of 24.2 milligrams and standard deviation of 2.7 milligrams for a sample of n = 9 cigarettes.

Since the sample size is small in your example, you would most likely need to use the t-distribution when constructing a confidence interval to account for the smaller sample size.

To determine if the sample data can be used to construct a confidence interval, you need to check if the sample meets the necessary assumptions and conditions. Here are the steps you can follow:

1. Identify the characteristics of the data: Determine if the data you have is quantitative or categorical. Confidence intervals are typically used for quantitative data.

2. Check if the sample is random: The data should come from a randomly selected sample to ensure that it is representative of the population you are interested in. If the sample was not selected randomly, it may not be appropriate to construct a confidence interval.

3. Determine the sample size: The sample size should be large enough so that the sampling distribution of the sample mean or proportion is approximately bell-shaped (following the Central Limit Theorem). For means, a sample size of at least 30 is often considered sufficient, while for proportions, it is recommended to have at least 10 successes and 10 failures in the sample.

4. Assess the independence assumption: The observations in the sample should be independent of each other. This means that one observation should not influence the next. If the observations are dependent, such as in a paired design or time series data, the independence assumption may not hold, and you may need to use specialized techniques.

5. Evaluate the distribution of the data: To construct a confidence interval, you typically assume that the sample data follows a normal distribution or that the sample size is large enough for the Central Limit Theorem to apply. You can check this assumption by examining the shape of the data using histograms, box plots, or normality tests like the Shapiro-Wilk or Anderson-Darling test.

6. Check for outliers or influential observations: Outliers or influential observations may affect the accuracy and reliability of the confidence interval. Identify any extreme values in the sample and assess if they significantly impact the results. If outliers are present, you may need to consider alternative methods or robust techniques.

After going through these steps, you can determine if the sample data satisfies the necessary assumptions and conditions for constructing a confidence interval. In the given example, you would need to assess if the sample of 9 cigarettes was randomly selected, if the sample size is appropriate, if the observations are independent, and if the nicotine content data follows a normal distribution or if the sample is large enough for the Central Limit Theorem to apply.