91Ó°ÊÓ

The article "Most Canadians Plan to Buy Treats, Many Will Buy Pumpkins, Decorations and/or Costumes" (Ipsos-Reid, October 24, 2005 ) summarized results from a survey of 1000 randomly selected Canadian residents. Each individual in the sample was asked how much he or she anticipated spending on Halloween during \(2005 .\) The resulting sample mean and standard deviation were \(\$ 46.65\) and \(\$ 83.70,\) respectively. a. Explain how it could be possible for the standard deviation of the anticipated Halloween expense to be larger than the mean anticipated expense. b. Is it reasonable to think that the distribution of the yariable anticipated Halloween expense is approximately normal? Explain why or why not. c. Is it appropriate to use the \(t\) confidence interval to estimate the mean anticipated Halloween expense for Canadian residents? Explain why or why not. d. If appropriate, construct and interpret a \(99 \%\) confidence interval for the mean anticipated Halloween exnense for Canadian residents.

Step by step solution

01

Understanding variance and mean

The variance or standard deviation in a data set, in this case \(\$ 83.70\), can be larger than the mean \(\$ 46.65\) if the data has several extreme values or if the data is spread out over a large range. It is possible that a few Canadian residents are anticipating to spend a lot on Halloween, and this increases the standard deviation.

02

Normality of distribution

Without more information or a histogram, it's not clear if the distribution is approximately normal. The significant difference between standard deviation and mean suggests that there might be outliers, which indicates that the data is not normally distributed. If many people are spending very little as compared to a few who are spending a lot, the data would be positively skewed.

03

Appropriateness of t-confidence interval

It's appropriate to use the t-confidence interval if the distribution is approximately normal or if the sample size is large. With a sample size of 1000, it's appropriate to use the t-confidence interval even if the data is not perfectly normal due to the Central Limit Theorem.

04

Construct a 99% confidence interval

For a 99% t-confidence interval with 999 degrees of freedom (1000 - 1), you refer to the t-distribution table to find that the t-score is close to 2.61. The formula for a confidence interval is \(\bar{X} ± (t * S/ \sqrt{n})\). Substituting the given values results in \$ 46.65 ± (2.61 * 83.70 / \sqrt{1000}) which gives \$ 46.65 ± 6.9. Therefore, the 99% confidence interval for the mean anticipated Halloween expense for Canadian residents is about \$ [39.75, 53.55] USD.

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Key Concepts

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

Standard Deviation

Standard deviation is a statistical measure representing the dispersion or variability within a data set. Basically, it tells us how spread out the numbers are around the mean (average). For instance, when considering the anticipated spending on Halloween, a high standard deviation, as we see with the \( \$ 83.70 \) figure, suggests that Canadians' spending habits are quite varied. Some may spend very little, whereas others might be splurging, leading to a broad range of expenditures.

A larger standard deviation relative to the mean, as in the given exercise, can indicate the presence of outliers or an asymmetric distribution of data. These factors can significantly affect the average, causing a mean that doesn't reflect the 'typical' value as accurately as we might presume when we see a smaller standard deviation.

Mean

The mean, often referred to as the average, is a central tendency measure that sums up the data set's overall pattern. It is calculated by adding up all the values and then dividing by the number of values. For the Halloween spending data, the mean anticipated expense is \( \$ 46.65 \). This is the average amount that the surveyed Canadian residents expect to spend.

The mean is sensitive to extreme scores, known as outliers, which can pull the mean in the direction of the outlier and may not accurately represent the center of the data. This sensitivity highlights the importance of considering the mean in conjunction with measures like standard deviation to get a fuller understanding of the data's characteristics.

T-Distribution

The t-distribution, sometimes known as Student's t-distribution, comes into play particularly when we're estimating the mean of a population and the sample size is small, or the population standard deviation is unknown. Unlike the normal distribution, the t-distribution is more likely to have values at the extremes (i.e., heavier tails), which compensates for the variability and uncertainty associated with smaller sample sizes.

In the context of the Halloween spending data, because the population standard deviation is unknown and we only have a sample standard deviation, the t-distribution provides a more accurate way to estimate the population mean. It becomes crucial for calculating confidence intervals when working with smaller samples, or in this case, with large samples but where normality can't be assumed due to extreme values.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that states that the distribution of sample means will approximate a normal distribution, regardless of the shape of the original distribution, as the sample size becomes large. The theorem provides a foundation for making inferences about a population from sample data.

In relation to the Halloween spending data, even if the original distribution of spending is skewed or not normal due to the presence of outliers or a high standard deviation, the CLT allows us to use the normal (or t-) distribution to make inferences about the mean spending because the sample size is large (\(n = 1000\)). This large sample size gives us confidence that the sampling distribution of the mean will be approximately normal, which is why a t-confidence interval can be used.

Sample Size

Sample size matters immensely in statistics. It refers to the number of observations or data points collected in a study. The larger the sample size, the more confidence statisticians can have in their estimates and the smaller the margin of error. With a large enough sample, even non-normal data can produce a normal distribution of sample means, as per the Central Limit Theorem.

In our example with the Halloween data, the sample size is 1000, which is quite substantial. This large sample size provides a solid basis for estimating the mean anticipated Halloween expense and calculating the confidence interval, giving us a strong level of certainty about our estimates.

Normality of Distribution

The assumption of normality is central to many statistical procedures and theories. A normal distribution is a symmetric, bell-shaped curve where most of the data points are clustered around the mean, with probabilities for values tapering off as they go further away from the mean.

In practice, not all data sets follow a normal distribution, and this can be inferred from the relationship between the mean and the standard deviation. If the latter is significantly larger than the former, as is the case with the anticipated Halloween expenses (\( mean = \$ 46.65 \) and \( standard deviation = \$ 83.70 \) ), it suggests that the distribution may be skewed or have outliers. This is why assessing the normality of the distribution is crucial before applying certain statistical tests that assume normality, although, with large samples, the Central Limit Theorem ensures that the means of samples will tend to a normal distribution.