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In mid-2016 the United Kingdom (UK) withdrew from the European Union (an event known as "Brexit"), causing economic concerns throughout the world. One indicator that economists use to monitor the health of the economy is the proportion of residential properties offered for sale at auction that are successfully sold. An article titled "Going, going, gone through the roof-sky's the limit at auction" (October \(22,2016,\) www.estateagenttoday .co.uk/features/2016/10/going-going-gone-through-theroof-the-skys-the-limit- at-auction, retrieved May 4,2017 ) reported the success rate of a sample of 26 residential properties offered for sale at auctions in the UK in the summer of \(2016 .\) For this sample of properties, 14 of the 26 residential properties were successfully sold. Suppose it is reasonable to consider these 26 properties as representative of residential properties offered at auction in the post-Brexit UK. a. Would it be appropriate to use the large-sample confidence interval for a population proportion to estimate the proportion of residential properties successfully sold at auction in the post-Brexit UK? Explain. b. Would it be appropriate to use a bootstrap confidence interval for a population proportion to estimate the proportion of residential properties successfully sold at auction in the post-Brexit UK? Explain. c. Use the accompanying output from the "Bootstrap Confidence Interval for One Proportion" Shiny app to report a \(95 \%\) bootstrap confidence interval for the population proportion of residential properties successfully

Step by step solution

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a. Large-Sample Confidence Interval Appropriateness

To determine if it is appropriate to use a large-sample confidence interval, we need to check if the sample size is large enough. A rule of thumb states that both the number of successes (n*p) and failures (n*(1-p)) should be greater than or equal to 10. Here, n=26 (total number of properties) and p=14/26 (success proportion). Let's check if the conditions are met: \(n*p = 26*(14/26) = 14\) \(n*(1-p) = 26*(1-14/26) = 12\) Both values are greater than or equal to 10, so it would be appropriate to use the large-sample confidence interval for estimating the population proportion.

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b. Bootstrap Confidence Interval Appropriateness

The bootstrap method is a non-parametric approach that does not assume any specific distribution of the population. It is suitable for small samples or for cases where the distribution is unknown. Since the sample size in this situation is not large, it would be appropriate to use a bootstrap confidence interval for estimating the population proportion of residential properties successfully sold at auction in the post-Brexit UK.

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c. 95% Bootstrap Confidence Interval Calculation

We will use the Shiny app mentioned in the exercise to find the 95% confidence interval for the population proportion. 1. Open the Shiny app and select "Bootstrap Confidence Interval for One Proportion". 2. Input the sample size (26) and the number of successes (14) in the respective input boxes. 3. Choose a confidence level of 95%. The app will provide the bootstrap confidence interval for the population proportion. Report the confidence interval obtained from the app.

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Key Concepts

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

Large-Sample Confidence Interval

The concept of large-sample confidence interval is fundamental in inferential statistics, particularly when estimating a population parameter, such as a proportion. This method is grounded on the Central Limit Theorem which posits that, for a large enough sample size, the distribution of sample proportions approximates a normal distribution, regardless of the population's distribution.

When dealing with large samples, typically those with at least 30 observations, statisticians can apply the large-sample confidence interval technique. The premise is to use the sample proportion to infer about the population proportion within a specified level of confidence, usually 95% or 99%. It's crucial to check that both success and failure occurrences within the sample are adequate, generally each exceeding 10, to ensure reliable interval estimation. In the context of residential property sales, as in our exercise, these checks help confirm whether the large-sample interval can accurately reflect the population's behavior post-Brexit.

Bootstrap Confidence Interval

In cases where traditional assumptions for large-sample confidence intervals are not met, or when the sample size is small, the bootstrap confidence interval comes into play. Bootstrap is a powerful non-parametric method that resamples the original data with replacement to create 'bootstrap samples'. These samples are then used to calculate the statistic of interest multiple times, resulting in an empirical distribution.

This approach doesn't rely on the large-sample theory or any assumptions about the underlying population distribution. That makes it particularly useful for small or asymmetrical datasets, as well as for complex estimands where standard formulae for confidence intervals are not applicable. With regards to estimating the population proportion of UK residential properties sold at auction post-Brexit, the bootstrap method can provide a robust confidence interval even from a small sample of 26 properties.

Population Proportion

The term population proportion signifies the fraction of a population that possesses a particular attribute. It is a kind of parameter that statisticians estimate to generalize about a broader group based on a sample. For instance, in the context of our exercise, the population proportion of interest is the proportion of residential properties sold at auction post-Brexit.

To estimate this proportion, we use the sample of 26 properties - where 14 were sold - as a snapshot to make inferences about the entire market. In doing so, estimating the population proportion involves calculating the sample proportion (here, 14 out of 26) and then constructing a confidence interval around it to express the uncertainty inherent in the estimation process. Statistical methods, including large-sample and bootstrap confidence intervals, serve to provide this estimation with varying degrees of assumption and robustness.

Non-Parametric Methods

Finally, let's delve into non-parametric methods, a class of statistical techniques that do not assume a specific statistical distribution. These methods are valuable when dealing with small samples or when the data does not conform to normal distribution assumptions that underpin many parametric tests.

Non-parametric methods, such as the bootstrap technique discussed earlier, use the data's actual distribution to perform calculations. They are flexible and adapt to the data's inherent characteristics, thus providing more accurate results under certain conditions. The bootstrap method for confidence interval estimation exemplifies a non-parametric approach, offering a viable alternative to traditional parametric methods, particularly suitable for our case of evaluating the post-Brexit UK residential property market.