Sampling Distributions The following data represent the running lengths (in minutes) of the winners of the Academy Award for Best Picture for the years 2012–2017.

f. Repeat parts (b)–(e) using samples of size n=3. Comment on the effect of increasing the sample size.

Verified step by step guidance

1. Step 1: Identify all possible samples of size \(n=3\) from the given data set of running times: 120, 134, 119, 129, 111, and 123 minutes. Since the sample size is 3, list all combinations of 3 running times without repetition.
2. Step 2: For each sample of size 3, calculate the sample mean using the formula: \(\text{Sample Mean} = \frac{X_1 + X_2 + X_3}{3}\), where \(X_1, X_2, X_3\) are the running times in the sample.
3. Step 3: Create the sampling distribution of the sample mean by listing all the sample means calculated in Step 2. This distribution shows the means of all possible samples of size 3.
4. Step 4: Calculate the mean and standard deviation of the sampling distribution of the sample mean. The mean of the sampling distribution should be close to the population mean, and the standard deviation (standard error) can be calculated using \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n=3\).
5. Step 5: Compare the sampling distribution for \(n=3\) with the previous sampling distribution for smaller sample sizes (e.g., \(n=2\)). Comment on how increasing the sample size affects the variability (standard error) and the shape of the sampling distribution, noting that larger sample sizes tend to produce sampling distributions with smaller variability and more normal shape.

Key Concepts

Sampling Distribution

A sampling distribution is the probability distribution of a given statistic based on a random sample. It shows how the statistic varies from sample to sample and is fundamental for making inferences about a population from sample data.

Sample Size and Its Effect

Sample size (n) refers to the number of observations in a sample. Increasing the sample size generally reduces the variability of the sampling distribution, leading to more precise estimates of population parameters.

Central Limit Theorem

The Central Limit Theorem states that, for sufficiently large sample sizes, the sampling distribution of the sample mean approaches a normal distribution regardless of the population's distribution. This allows for easier inference using normal probability models.