A new Web page design is intended to increase the rate at...

A new Web page design is intended to increase the rate at which customers place orders Prior to the new design, the number of orders in an hour was a Poisson random variable with mean 30 . Eight one-hour measurements with the new design find an average of 32 orders completed per hour. (a) At a 5\% significance level, do the data support the claim that the order placement rate has increased? (b) Repeat part a at a $1 \%$ significance level.

A new Web page design is intended to increase the rate at which customers place orders Prior to the new design, the number of orders in an hour was a Poisson random variable with mean 30 . Eight one-hour measurements with the new design find an average of 32 orders completed per hour.
(a) At a 5\% significance level, do the data support the claim that the order placement rate has increased?
(b) Repeat part a at a $1 \%$ significance level.

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution used to model the number of times an event occurs in a fixed interval of time or space. It is characterized by its mean (?), which represents both the average rate of occurrence and the variance. This distribution is particularly useful when dealing with count data that occur independently and sporadically over time.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves formulating two competing hypotheses (the null hypothesis and the alternative hypothesis) and then using data to assess the strength of evidence against the null hypothesis. This process aids in determining whether an observed effect is statistically significant.

Null and Alternative Hypotheses

In the context of statistical testing, the null hypothesis (H?) generally represents the status quo or a position of no change or no effect, while the alternative hypothesis (H?) represents the claim being tested, often indicating a difference or an increase in a parameter. Correct formulation of these hypotheses is crucial for appropriate analysis and interpretation of results.

Significance Level

The significance level, denoted by ?, is the threshold probability for rejecting the null hypothesis. It defines the maximum acceptable probability of committing a Type I error, which is the error of rejecting a true null hypothesis. Common significance levels are 5% and 1%, and they set the criteria for decision-making in hypothesis testing.

P-value

The p-value is a measure used in hypothesis testing to quantify the strength of the evidence against the null hypothesis. It represents the probability of obtaining a result at least as extreme as the observed sample data, assuming that the null hypothesis is true. A smaller p-value indicates stronger evidence in favor of the alternative hypothesis.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental statistical principle stating that the distribution of the sample mean approximates a normal distribution as the sample size becomes large, regardless of the original distribution of the data. This concept is especially important when analyzing sample means from Poisson distributions, as it allows for the use of normal approximation methods in hypothesis testing.

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