Question

A plant manager claims that on the average no more than 5 service calls per hour are made by the plant's workers. Suppose in one particular hour, 9 service calls were required. At a \(5 \%\) level of significance, could we now reject the plant manager's claim?

Short answer

Based on the information provided, we are unable to perform hypothesis testing with a one-sample t-test due to having only one data point (9 service calls in 1 hour). To make a valid conclusion, we would need more data on service calls for a larger number of hours, calculate the sample mean and standard deviation, and then perform a hypothesis test to determine whether or not we could reject the plant manager's claim at a 5% level of significance.

Step-by-step solution

State the Null and Alternative Hypotheses

In hypothesis testing, we need to define our null hypothesis and alternative hypothesis. The null hypothesis is the plant manager's claim: no more than 5 service calls per hour on average. The alternative hypothesis is that there are more than 5 service calls per hour on average. Null hypothesis (H0): \(\mu \leq 5\) Altеrnative hypothesis (H1): \(\mu > 5\)

Determine the Significance Level

The problem states that we'll be using a 5% level of significance, so our alpha, α, will be: α = 0.05

Calculate the Test Statistic (t-score)

Since we only have one sample of 1 hour, where 9 service calls were required, we are unable to actually calculate the t-score. To perform a reliable hypothesis test, we would need to have a larger sample size (preferably at least 30) and calculate the sample mean and sample standard deviation in order to find the t-score, which would then be compared to the critical value (t-critical value) to determine whether the null hypothesis can be rejected or not.

Conclusion

With only one data point (9 service calls in 1 hour), we do not have enough information to perform hypothesis testing with a one-sample t-test. Thus, we are unable to conclude whether we can reject the plant manager's claim at a 5% level of significance. In order to make a valid conclusion, we would need to gather more information about the service calls in that plant for a larger number of hours, calculate the sample mean and standard deviation, and perform a hypothesis test with a larger sample size.

Key concepts

Null Hypothesis

In hypothesis testing, we start by identifying the **Null Hypothesis** (commonly denoted as **H0**). This acts as the default or the initial claim we aim to test. In the given exercise, the null hypothesis represents the plant manager's assertion, which states that no more than 5 service calls are made per hour on average. In symbolic form, it is written as

• \(H_0: \mu \leq 5\)

This means that the average number of service calls per hour is 5 or fewer. The role of the null hypothesis is crucial as it forms the basis of statistical testing, guiding us in deciding whether there is enough evidence to reject it, thereby challenging the original claim.
When conducting hypothesis testing, the null hypothesis is assumed to be true until evidence suggests otherwise. Remember, not rejecting the null hypothesis doesn’t prove it's true; it simply means that there isn't sufficient evidence to conclude it's false.

Alternative Hypothesis

The **Alternative Hypothesis** challenges the null hypothesis by suggesting that something different from the initial claim is happening. It proposes a change or effect, indicating that what we are testing holds enough ground to be considered true. In our exercise, the alternative hypothesis is that more than 5 service calls are made on average per hour. This is expressed mathematically as:

• \(H_1: \mu > 5\)

The alternative hypothesis is critical as it outlines what we hope to support through our data: that the number of service calls exceeds the claimed average.
The alternative hypothesis is crucial for determining the direction of our test. Here, it's a one-tailed test because it specifies the direction of the greater number of calls. Ultimately, the goal of hypothesis testing is to evaluate whether the alternative hypothesis provides a better explanation given the data, leading either to the rejection of the null hypothesis or failing to do so.

Significance Level

The **Significance Level** (denoted by \(\alpha\)) is a threshold we set to determine how extreme our data must be before we can reject the null hypothesis. In simpler terms, it quantifies the probability of making a Type I error – rejecting the null hypothesis when in fact it is true.

• In our exercise, the significance level is set at 5%, or \(\alpha = 0.05\).

This means we accept a 5% risk of assuming a difference or effect exists when it does not. The choice of significance level often reflects the rigor or confidence desired in the test results.
With a 5% significance level, you're saying you require 95% confidence in your results before you proclaim there's sufficient evidence against the null hypothesis. Throughout statistical testing, a lower \(\alpha\) often demands more convincing evidence from the data to support any claims, ensuring fewer false-positive conclusions.

Test Statistic

The **Test Statistic** is a standardized value calculated from sample data during a hypothesis test. It is used to decide whether to reject the null hypothesis. In the context of hypothesis testing, it bridges our data with the underlying probability distribution to evaluate the plausibility of the null hypothesis.
In our case, using a one-sample t-test is advised because we have a specific value for the average number of calls claimed. However, given that only one sample is considered – 9 service calls in one hour – calculating the t-score isn't feasible. Ideally, the t-score would require a larger sample size, preferably greater than 30, along with calculated sample mean and standard deviation.
Once these calculations are possible, the test statistic is compared to a critical value derived from the relevant statistical distribution (like the t-distribution for small samples) to make an informed decision. This comparison helps to ascertain whether the observed data is significantly unusual under the null hypothesis, thus guiding us in either rejecting it or failing to do so.