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According to the website http://www.digitalbookworld.com, the average price of a bestselling ebook increased to \(\$ 8.05\) in the week of February 18,2015 from \(\$ 6.89\) in the previous week. Assume the standard deviation of the price of a bestselling ebook is \(\$ 1\) and suppose you have a sample of 20 bestselling ebooks with a sample mean of \(\$ 7.80\) and a standard deviation of \(\$ 0.95\) a. Identify the random variable \(X\) in this study. Indicate whether it is quantitative or categorical. b. Describe the center and variability of the population distribution. What would you predict as the shape of the population distribution? Explain. c. Describe the center and variability of the data distribution. What would you predict as the shape of the data distribution? Explain. d. Describe the center and variability of the sampling distribution of the sample mean for 20 bestselling ebooks. What would you predict as the shape of the sampling distribution? Explain.

Step by step solution

01

Understand Random Variables

The random variable in this context is the price of a bestselling ebook. Since prices are numerical values, this random variable is quantitative.

02

Population Distribution Analysis

The average price (mean) of the population distribution is given as \( \mu = 8.05 \). The standard deviation of the population distribution is \( \sigma = 1.00 \). Without information about skewness or other distributional characteristics, we might expect the population distribution to be approximately normal due to the nature of pricing data, but it is difficult to explicitly determine the shape without more data.

03

Data Distribution Analysis

The sample mean (center of data distribution) is \( \bar{x} = 7.80 \), and the sample standard deviation is \( s = 0.95 \). The sample size is small (20), so without explicit indication of normality, the data distribution shape is indeterminate but might be close to normal due to central tendencies around the mean.

04

Sampling Distribution of the Sample Mean

For the sample mean distribution, the central limit theorem suggests that it will be approximately normal, even with the small sample size of 20. The mean of the sampling distribution is the same as the population mean \( \mu = 8.05 \), and the standard deviation (standard error) is calculated using \( \sigma / \sqrt{n} = 1 / \sqrt{20} \approx 0.2236 \). Thus, we predict the shape of this distribution to be approximately normal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Variables

In statistics, understanding the concept of random variables is crucial. A random variable is a numerical outcome of a random event or phenomenon. In simple terms, it's the value that we measure or observe, and it can vary from one event to another. In the context of our ebook pricing exercise, the random variable is the price of a bestselling ebook. This is a perfect example of a quantitative random variable because it can take on any value within a range and is numerical in nature.

• Quantitative random variables are always numerical.
• They can be either discrete (countable values) or continuous (any value within a range).

In our case, the price of ebooks could theoretically take on any value, making it continuous. Understanding whether a random variable is quantitative or categorical helps in determining how data should be analyzed and interpreted.

Sampling Distribution

The sampling distribution is a powerful concept that helps us understand how sample statistics behave. In our ebook example, if we were to take several samples of ebook prices, each sample would yield its own mean. The collection of all those sample means forms what we call the sampling distribution. It’s a big-picture view of how a particular statistic (like the mean) behaves over many samples.

• The mean of the sampling distribution is expected to be the same as the mean of the population distribution.
• The shape of the sampling distribution tends to be normal, especially as sample size grows.
• We calculate the standard deviation of the sampling distribution, known as the standard error, using the formula: \( \text{SE} = \sigma / \sqrt{n} \).

For our sample of 20 ebooks, this results in a standard error of 0.2236. This helps analysts understand how much sample means deviate from the actual population mean. It's a foundation for making predictions about how estimates from samples relate to entire populations.

Central Limit Theorem

The Central Limit Theorem (CLT) is one of the cornerstones of statistics. It tells us that, regardless of the shape of a population distribution, the sampling distribution of the sample mean will be approximately normal, provided we have a sufficiently large sample size. This theory allows statisticians to make inferences about population parameters even when the population distribution itself is not normal.

• The CLT holds true particularly well for sample sizes of 30 or more.
• Even for smaller samples, like our 20 exmaple, the sampling distribution is often roughly normal, as supported by the CLT.

The standard error becomes crucial here because it quantifies how much the sample mean is expected to vary from the true population mean. Through the power of the CLT, we gain great flexibility when analyzing data and predicting outcomes, as statistical techniques assuming normality can still be applied.

Population Distribution Analysis

When we talk about population distribution, we refer to the range and frequency of a characteristic within an entire population. For our ebook price example, the population distribution is characterized by its mean, \( \mu = 8.05 \), and standard deviation, \( \sigma = 1.0 \). These values provide important details about the overall trend and variability in the ebook prices.

• A higher standard deviation implies more variability in prices.
• Without additional data, we often assume a normal distribution for population prices because of central tendencies.
• Prediction about the shape of the population distribution remains speculative without more descriptors—skewness or kurtosis, for instance.

Analyzing the population distribution helps us understand the spread and tendency of prices across all bestsellers, enabling better decision-making and estimation in business strategies.