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In the world of probability distributions, the mean and standard deviation are crucial concepts to grasp. The mean, denoted by \( \mu \), represents the expected or average value of a probability distribution. In this lottery scenario, \( \mu = 0.10 \), which means that on average, a player expects to win 10 cents per bet. This small mean reflects the high odds against guessing the right six-digit number.
The standard deviation, represented by \( \sigma \), measures the spread of the data around the mean. It tells us how much the winnings varied from the average. With a standard deviation of \( \sigma = 100.00 \), there's significant variability. This large standard deviation is due to the lottery's structure, where a rare win delivers a substantial payout of \(100,000, but most outcomes result in a loss of \)1.
To summarize:

• Mean (\( \mu \)): Expected average outcome, which is 10 cents here.
• Standard Deviation (\( \sigma \)): Variability from the mean, at a considerable $100 due to the nature of the game.

Understanding sampling distributions is key when determining parameters like the mean and standard deviation of sample data, especially over a large number of trials. A sampling distribution of the sample mean represents the distribution of all possible sample means that could be drawn from the population. This is particularly relevant when Joe plays the lottery 1 million times, creating a wide array of possible outcomes for each of his attempts.
When dealing with a sample mean \( \bar{x} \), the mean of its sampling distribution, denoted \( \mu_{\bar{x}} \), remains the same as the population mean, \( \mu \). Therefore, \( \mu_{\bar{x}} = 0.10 \), providing an expected average outcome across all of Joe's plays.
Moreover, the standard deviation of the sampling distribution (known as the standard error), differs. It is calculated using the formula \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \). For Joe's 1 million plays, the value is greatly reduced to \( \sigma_{\bar{x}} = 0.10 \). This smaller figure highlights increased precision in the sample mean's estimate of the population mean as more samples are drawn.

• Mean of Sampling Distribution (\( \mu_{\bar{x}} \)): Same as the population mean (10 cents).
• Standard Error (\( \sigma_{\bar{x}} \)): Lower at 0.10, offering a more precise estimate.

The Central Limit Theorem (CLT) is a cornerstone of statistics, helping us understand why Joe's average winnings follow a normal distribution when he plays the lottery many times. According to the CLT, regardless of the original distribution (which can be very skewed or varied), the sampling distribution of the sample mean tends to be approximately normal if the sample size is sufficiently large.
In this problem, Joe plays the lottery 1 million times—well beyond a large enough sample size for the CLT to apply. As a result, his average winnings \( \bar{x} \) assume a bell-shaped curve, even though his individual outcomes per bet vary dramatically.
This theorem allows us to calculate probabilities for Joe's average winnings. For instance, to find the probability Joe exceeds \(1 on average (when he wagers \)1 per play), we use the z-score obtained from normal distribution tables. With a z-score of 9, indicating many standard deviations away from the mean, it's exceedingly unlikely Joe's average will surpass $1.
Key takeaways include:

• CLT Assumption: Sampling distribution of the mean becomes normal with large samples.
• Normal Distribution Application: Facilitates probability calculations for large samples.