Question

In a report of why e-shoppers abandon their online sales transactions, Alison Stein Wellner \(^{8}\) found that "pages took too long to load" and "site was so confusing that I couldn't find the product" were the two complaints heard most often. Based on customers' responses, the average time to complete an online order form will take 4.5 minutes. Suppose that \(n=50\) customers responded and that the standard deviation of the time to complete an online order is 2.7 minutes. a. Do you think that \(x\), the time to complete the online order form, has a mound-shaped distribution? If not, what shape would you expect? b. If the distribution of the completion times is not normal, you can still use the standard normal distribution to construct a confidence interval for \(\mu\), the mean completion time for online shoppers. Why? c. Construct a \(95 \%\) confidence interval for \(\mu,\) the mean completion time for online orders.

Short answer

Also, what would be the 95% confidence interval for the mean completion time? Answer: We cannot determine the distribution shape of completion times from the given information. However, we can still use the standard normal distribution to construct a confidence interval for the mean because of the Central Limit Theorem (CLT). When the sample size is sufficiently large (n > 30), the sampling distribution of the sample mean approaches normality, regardless of the shape of the population distribution. In this case, our sample size is \(n = 50\), which is greater than 30. The 95% confidence interval for the mean completion time lies between 3.753 and 5.247 minutes.

Step-by-step solution

a. Distribution Shape

To determine the distribution shape of completion times, we need more information than just the average and standard deviation. If we were given the skewness or kurtosis values, we could have made better guesses. However, based on given information, it is not possible to determine the shape of the distribution accurately.

b. Using the Standard Normal Distribution

Even if the distribution of completion times is not normal, we can still use the standard normal distribution to construct a confidence interval for the mean. This is because of the Central Limit Theorem (CLT). According to CLT, if we have a sample size that is sufficiently large (generally \(n > 30\)), the sampling distribution of the sample mean approaches normality, regardless of the shape of the population distribution. In this case, our sample size is \(n = 50\), which is greater than 30, so we can use the standard normal distribution to construct the confidence interval for the mean completion time.

c. Constructing a 95% Confidence Interval for the Mean

Given the average time (\(\overline{x} = 4.5\) minutes), the standard deviation of time (\(s = 2.7\) minutes), and a sample size (\(n = 50\)), we can calculate the standard error (SE) by using the following formula: \(SE = \frac{s}{\sqrt{n}}\) Plugging in the values: \(SE = \frac{2.7}{\sqrt{50}} \approx 0.381\) For a 95% confidence interval, we'll use the standard normal critical value (\(z^*\)) of 1.96 (from the z-table). Now we can calculate the margin of error (ME): \(ME = z^* \times SE = 1.96 \times 0.381 \approx 0.747\) Finally, the 95% confidence interval for the mean completion time is: \((\overline{x} - ME, \overline{x} + ME) = (4.5 - 0.747, 4.5 + 0.747) = (3.753, 5.247)\) Thus, we can conclude with 95% confidence that the mean time to complete the online order form for all customers lies between 3.753 and 5.247 minutes.

Key concepts

Central Limit Theorem

Understanding the Central Limit Theorem (CLT) is crucial when it comes to making inferences about a population based on a sample. The theorem states that, given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. Moreover, the sampling distribution of the sample mean will approach a normal distribution regardless of the population's distribution shape.

For example, even if we're unsure about whether the distribution of online order completion times is perfectly mound-shaped, the CLT allows us to assume normality for the purpose of constructing confidence intervals. In our exercise, a sample size of 50 (\(n=50\)) is large enough since it's greater than the recommended threshold of 30. This is why we can safely use the normal distribution in subsequent calculations to estimate the population mean, \(\mu\), for online completion times.

It is important to understand that the CLT only applies when the sample size is large and does not suggest that the actual population distribution is normal. The 'magic' of CLT makes it possible for us to apply normal probability to a wide range of problems, providing a powerful tool for statisticians and data analysts.

Standard Error

The standard error (SE) is a measure that describes how far the sample mean (\(\bar{x}\)) is likely to be from the actual population mean (\(\mu\)). In other words, it gauges the precision of the sample mean as an estimate of the population mean. The lower the standard error, the closer we can expect our sample mean to be to the population mean, thereby indicating a more precise estimate.

The formula to calculate the standard error is:
\[SE = \frac{s}{\sqrt{n}}\]
Where \(s\) is the standard deviation of the sample, and \(n\) is the sample size. In the context of our exercise, with a sample standard deviation of 2.7 minutes and a sample size of 50, the calculated standard error is approximately 0.381 minutes. This small standard error suggests that the sample mean of our e-shopper study is a good estimator of the true mean completion time for all online shoppers.

Understanding standard error is vital for constructing confidence intervals as it directly impacts the margin of error, enabling us to describe the range within which we can expect the population mean to fall. This relationship highlights the importance of both sample size and variability in estimating population parameters.

Standard Normal Distribution

The standard normal distribution is a specific type of normal distribution with a mean of zero (0) and a standard deviation of one (1). It is a critical concept in statistics because it allows for the simplification of complex probability problems. Representing the normal distribution in a standardized form makes it easier to calculate probabilities for any normal distribution.

Calculating a confidence interval often involves using the standard normal distribution, which is the basis for the 'z-table', a table that provides areas under the normal curve for various z-value cutoffs. In the exercise solution, the z-value associated with a 95% confidence level is 1.96. This represents the number of standard deviations from the mean required to encompass the middle 95% of the data in a standard normal distribution.

To compute the margin of error (ME) in the construction of a confidence interval, this z-value is multiplied by the standard error. This results in a value that determines how far from the sample mean the confidence interval extends on either side. Because the standard normal distribution is symmetrical, we can easily derive the lower and upper bounds for the interval in which we believe the population mean is likely to lie. In our exercise, this leads to a 95% confidence interval for the mean time to complete an online order form between approximately 3.753 and 5.247 minutes.

The advantage of the standard normal distribution is that it provides a common reference for comparing statistical measures across different scenarios, and it plays a foundational role in many statistical methods, including hypothesis testing and constructing confidence intervals.