Question

The article "Portable MP3 Player Ownership Reaches New High" (Ipsos Insight, June 29,2006 ) reported that in \(2006,20 \%\) of those in a random sample of 1112 Americans age 12 and older indicated that they owned an MP3 player. In a similar survey conducted in 2005 , only \(15 \%\) reported owning an MP3 player. Suppose that the 2005 figure was also based on a random sample of size 1112 . Estimate the difference in the proportion of Americans age 12 and older who owned an MP3 player in 2006 and the corresponding proportion for 2005 using a 95\% confidence interval. Is zero included in the interval? What does this tell you about the change in this proportion from 2005 to \(2006 ?\)

Short answer

To provide a precise short answer, the standard error - SE needs to be calculated first. After that one could state the confidence interval and whether zero was included in it or not, and explain what the implications of that are.

Step-by-step solution

Calculation of Sample Proportions

The number of participants who reported owning an MP3 player in each year can be calculated by multiplying the total sample size (1112) by the respective proportions. For 2005, the number is \(1112 \times 0.15 = 167\) and for 2006, it is \(1112 \times 0.20 = 222\).

Estimation of the Difference in Proportions

The difference in the reported proportions between 2005 and 2006 can be estimated as \( p_{2006} - p_{2005} = 0.20 - 0.15 = 0.05\).

Calculation of Standard Error

The standard deviation of the difference (standard error) can be estimated using the formula \( SE = \sqrt{ \frac{p_{2005} \times (1 - p_{2005})}{n_{2005}} + \frac{p_{2006} \times (1 - p_{2006})}{n_{2006}} } \). Substituting with the given values, we get \( SE = \sqrt{ \frac{0.15 \times 0.85}{1112} + \frac{0.20 \times 0.80}{1112} } \).

Calculation of the 95% Confidence Interval

A 95% confidence interval for the difference in proportions can be calculated as \( (p_{2006} - p_{2005}) \pm Z \times SE \), where Z = 1.96, which is the Z score for a 95% confidence interval. Substituting with the estimated values from the earlier steps, we get \( CI = 0.05 \pm 1.96 \times SE \).

Interpretation of the Confidence Interval

If zero is not included in the confidence interval, it means that there was a significant change in the proportion of Americans owning MP3 players from 2005 to 2006. If zero is included in the confidence interval, it means that the change was not significant.

Key concepts

Statistics

In the context of the given exercise, statistics provide us with methods and tools to analyze and interpret data. The process begins with data collection, which in this case is the survey of American MP3 player ownership over two different years.

Statistical analysis then enables us to draw conclusions from this data, such as estimating proportions and comparing them across groups or over time. By using statistical techniques, we can infer population trends or preferences based on sample data, such as determining if the popularity of MP3 players has significantly increased from one year to the next. A 95% confidence interval, as used in this exercise, is a key statistical concept which gives us a range in which we can say, with a certain level of confidence, that the true difference in population proportions lies.

If this interval does not include the value zero, we infer that there is a statistically significant difference, indicating a real change in MP3 player ownership between the years studied. Understanding this concept underscores the importance of statistical knowledge in interpreting survey results and making informed decisions or predictions.

Proportion Estimation

Proportion estimation is an essential area of statistics when we deal with categorical data. It involves estimating the proportion of a population that has a specific characteristic, based on sample data. In our exercise example, we estimate the proportion of Americans age 12 and older who owned an MP3 player based on a sample.

For accurate proportion estimation, the sample must be representative of the population, which is why a random sample is critical. The steps provided in the solution detail calculating the sample proportions and using these to estimate differences. By calculating a confidence interval around this estimated difference, we can measure the reliability of the estimation.

The narrower the confidence interval, the more precise our estimation is considered to be. This precision is influenced by the sample size and the variability within the data. Hence, understanding the principles behind proportion estimation is vital for interpreting and validating the results of data analysis, especially in surveys and studies that use sampling methods.

Data Analysis

Data analysis is the process of systematically applying statistical or logical techniques to describe and illustrate, condense and recap, and evaluate data. In the setting of our problem, data analysis involves estimating population parameters such as the ownership of MP3 players, comparing those estimates across different time points, and determining the significance of the observed changes.

The step-by-step solution we have guides us through calculating the estimated proportion of owners in each year, evaluating the standard error of the difference in proportions (which is a measure of uncertainty in our estimation), and constructing a confidence interval. Analyzing this interval helps us determine whether the difference observed in the sample is likely reflective of a real difference in the wider population.

Data analysis is integral in various fields, not just in statistics but also in business, science, and more. Mastery of data analysis techniques leads to better decision-making based on empirical evidence. In educational contexts, enhancing students’ understanding of data analysis equips them with the ability to critically evaluate and derive meaning from data which is ubiquitous in today’s information-driven world.