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The sign test is a very simple procedure for testing hypotheses about a population median assuming only that the underlying distribution is continuous. To illustrate, consider the following sample of 20 observations on component lifetime (hr): $$ \begin{array}{rrrrrrr} 1.7 & 3.3 & 5.1 & 6.9 & 12.6 & 14.4 & 16.4 \\ 24.6 & 26.0 & 26.5 & 32.1 & 37.4 & 40.1 & 40.5 \\ 41.5 & 72.4 & 80.1 & 86.4 & 87.5 & 100.2 & \end{array} $$ We wish to test \(H_{0}: \tilde{\mu}=25.0\) versus \(H_{\mathrm{a}}: \tilde{\mu}>25.0\). The test statistic is \(Y=\) the number of observations that exceed \(25 .\) a. Consider rejecting \(H_{0}\) if \(Y \geq 15\). What is the value of \(\alpha\) (the probability of a type I error) for this test? [Hint: Think of a "success" as a lifetime that exceeds 25.0. Then \(Y\) is the number of successes in the sample.] What kind of a distribution does \(Y\) have when \(\tilde{\mu}=25.0\) ? b. What rejection region of the form \(Y \geq c\) specifies a test with a significance level as close to \(.05\) as possible? Use this region to carry out the test for the given data.

Step by step solution

01

Identify the Problem

We have a sample of 20 component lifetimes, and we want to test the hypothesis \(H_0: \tilde{\mu} = 25.0\) against \(H_a: \tilde{\mu} > 25.0\) using the sign test. The test statistic \(Y\) is the number of observations exceeding 25.0.

02

Define Success and Distribution

In this context, a 'success' is measured as an observation greater than 25.0. Since the population median \(\tilde{\mu} = 25.0\), for each observation, the probability of success is \(0.5\). \(Y\), the number of successes in our sample, follows a Binomial distribution \(Y \sim B(n=20, p=0.5)\).

03

Calculate \(\alpha\) for \(Y \geq 15\)

\(\alpha\) is the probability of observing \(Y \geq 15\) under \(H_0\). Calculate this using the cumulative distribution for a binomial random variable. The value of \(\alpha\) is given by \(P(Y \geq 15) = \sum_{y=15}^{20} \binom{20}{y} (0.5)^y (0.5)^{20-y}\).

04

Perform Binomial Probability Calculations

Calculate the required probabilities for \(y = 15, 16, ... , 20\) under the binomial distribution \(B(20, 0.5)\) using the binomial probability formula: \[ P(Y = y) = \binom{20}{y} (0.5)^y (0.5)^{20-y} \]Sum these probabilities to find \(\alpha\).

05

Determine Appropriate Rejection Region

To find a rejection region of the form \(Y \geq c\) with significance level as close to 0.05 as possible, find a cutoff point \(c\) such that \(P(Y \geq c)\) is nearest to 0.05. Calculate \(P(Y \geq c)\) for different \(c\) values until the achieved significance level \(\alpha\) is close to 0.05.

06

Perform the Hypothesis Test

Count the actual number of sample observations exceeding 25.0. Check if this count falls into the rejection region determined. If it does, reject \(H_0\); otherwise, do not reject \(H_0\).

07

Calculate and Interpret Results

With the sample data provided, calculate the actual number of values exceeding 25. Comparing this to \(c\), determine if it is in the rejection region.

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Key Concepts

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

Population Median

The concept of the population median is central to understanding the sign test and hypothesis testing in statistics. In simple terms, the median is the middle value in a sorted list of numbers. When a distribution is continuous, the median divides the population into two equal halves, with 50% of values falling below it and 50% above it.

The population median is commonly denoted as \( \tilde{\mu} \). It serves as a critical value in hypothesis testing, particularly when considering tests like the sign test. In the given problem, we test whether the population median \( \tilde{\mu} \) is greater than 25.0.

Understanding the median's role helps identify whether a sample is centered around the assumed median or if there is significant deviation. This can influence decisions on hypothesis rejection.

Hypothesis Testing

Hypothesis testing is a method used to decide if there is enough evidence to reject a suppositional statement or hypothesis about a population parameter, like the median. In this context, we often start with a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

For the exercise at hand, our null hypothesis is \(H_0: \tilde{\mu} = 25.0\), meaning the median component lifetime is 25 hours. The alternative hypothesis, \(H_a: \tilde{\mu} > 25.0\), proposes that the median is greater than 25 hours.

The sign test used here evaluates how many sample observations exceed the median of interest, using that number to decide whether to reject \(H_0\). This process is crucial for testing assumptions about populations, helping statisticians make informed conclusions.

Binomial Distribution

The binomial distribution plays a vital role in hypothesis testing, particularly in tests like the sign test.

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials. In simpler terms, it tells us the likelihood of getting a specific number of 'successes' (e.g., lifetimes exceeding 25 hours) out of a fixed number of samples with a known probability of success.

In the context of the exercise, each lifetime observation is a trial, with a 'success' being an observation that exceeds 25 hours. Assuming \( \tilde{\mu} = 25.0 \), the probability of success for each observation is 0.5, making it a fair trial. With 20 trials, the relevant distribution for \(Y\), the number of successes, is \(Y \sim B(n=20, p=0.5)\). Knowing this helps calculate the probability of any number of observations exceeding the median.

Type I Error

A Type I error is an important concept in hypothesis testing. It occurs when the null hypothesis \(H_0\) is true, but we mistakenly reject it. In other words, we say there is an effect or difference when there actually isn't one.

The probability of making a Type I error is denoted \(\alpha\), and it’s a critical part of setting the significance level for a test. In our exercise, \(\alpha\) is the probability of rejecting the hypothesis that the median equals 25 (\(H_0\)), when it actually does.

Properly monitoring this error rate helps ensure our results are reliable. For instance, setting \(\alpha\) too high increases the chance of a Type I error, leading to false conclusions. Therefore, calculations for \(Y \geq c\) need caution to strike a balance between reducing Type I errors and maintaining test power.