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Chapter 8: Problem 49

Let \(\mu\) denote the mean reaction time to a certain stimulus. For a large- sample \(z\) test of \(H_{0}: \mu=5\) versus \(H_{\mathrm{a}}: \mu>5\), find the \(P\)-value associated with each of the given values of the \(z\) test statistic. a. \(1.42\) b. \(.90\) c. \(1.96\) d. \(2.48\) e. \(-.11\)

Short Answer

Expert verified
P-values are a) 0.0778, b) 0.1841, c) 0.0250, d) 0.0066, e) 0.5438.

Step by step solution

01

Understand the Hypotheses and Test Statistic

We want to conduct a large-sample one-tailed \(z\) test for the mean reaction time \(\mu\), with the null hypothesis \(H_0: \mu = 5\) and the alternative hypothesis \(H_a: \mu > 5\). The given \(z\) values will help determine whether we reject \(H_0\) based on the \(P\)-value.
02

Determine the P-Value for the Z-Test

For a one-tailed test where \(H_a: \mu > 5\), the \(P\)-value is the probability that the standard normal random variable \(Z\) is greater than the observed \(z\) value. We find \(P(Z > z)\) for each \(z\) value using a standard normal distribution table or calculator.
03

Calculate P-Value for each Z-Statistic

- **a. \(z = 1.42\):** The \(P\)-value is \(P(Z > 1.42) \approx 0.0778\).- **b. \(z = 0.90\):** The \(P\)-value is \(P(Z > 0.90) \approx 0.1841\).- **c. \(z = 1.96\):** The \(P\)-value is \(P(Z > 1.96) \approx 0.0250\).- **d. \(z = 2.48\):** The \(P\)-value is \(P(Z > 2.48) \approx 0.0066\).- **e. \(z = -0.11\):** The \(P\)-value is \(P(Z > -0.11) \approx 0.5438\).
04

Interpret P-Values

For each case, the \(P\)-value represents the probability of observing a \(z\) value as extreme as or more extreme than the one obtained, assuming \(H_0\) is true. Small \(P\)-values suggest that the null hypothesis is unlikely, indicating stronger evidence against \(H_0\). Common significance levels are \(0.05\) or \(0.01\).

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Key Concepts

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

P-Value
The P-Value is a crucial concept in statistical hypothesis testing. It helps us quantify the evidence against a null hypothesis. In essence, the P-Value tells us the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming the null hypothesis is true. In our example, we're conducting a Z-Test to decide if the mean reaction time, \( \mu \), is greater than 5 seconds. Each Z statistic given (like 1.42 or 0.90) corresponds to a specific P-Value.
  • If a P-Value is small (commonly less than 0.05 or 0.01), it suggests that the observed result would be very unlikely under the null hypothesis. Thus, providing strong evidence against \( H_0 \).
  • Larger P-Values imply that the observed data is much more consistent with \( H_0 \). Therefore, we lack evidence to reject the null hypothesis.
Calculating P-Values involves using the standard normal distribution, where we've already converted our data into Z statistics.
Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions based on data. It begins with two opposing hypotheses: the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \).
  • The null hypothesis (\(H_0\)) represents a statement of no effect or status quo. In our given problem, it asserts that the mean reaction time is 5 seconds.
  • The alternative hypothesis (\(H_a\)) is what we aim to provide evidence for. Here, it suggests the mean reaction time is greater than 5 seconds.
In this approach, one considers the P-Value to determine if the evidence is strong enough to reject the null hypothesis in favor of the alternative. If, based on the P-Value and a pre-set significance level (like 0.05), we reject \( H_0 \), it implies there's sufficient evidence to support \( H_a \). However, not rejecting \( H_0 \) doesn't necessarily mean it's true, only that we don't have enough evidence to prove otherwise.
Standard Normal Distribution
The standard normal distribution is fundamental to understanding Z-Tests and P-Values. This distribution is a special type of normal distribution that has a mean of 0 and a standard deviation of 1. It is represented by the symbol \( Z \) and used in calculating probabilities for hypothesis testing.
  • When conducting a test, raw data is often transformed into a Z score, which tells us how many standard deviations an element is from the mean. The Z score allows us to use the standard normal distribution table to find probabilities.
  • The area's lenient left and right of any Z value help determine the probabilities of events, which are crucial for finding P-Values. For instance, a Z score of 1.42 would have a certain probability associated, representing the P-Value, showing how significant this deviation is from the mean under \( H_0 \).
Hence, understanding this distribution enables us to carry out reliable hypothesis tests by assessing how extreme observed results are under the assumption that the null hypothesis is true.
One-Tailed Test
A one-tailed test in hypothesis testing examines if a parameter is distinct from the null hypothesis in only one direction. In our scenario, we're interested in whether the mean reaction time exceeds a set value (5), not whether it deviates in either direction.
  • This type of test increases power to detect an effect in one particular direction, as it doesn鈥檛 consider deviations in the opposite direction. This makes it especially useful when you're confident about the direction of your effect.
  • In a one-tailed Z-Test, the relevance of P-Values is clear. Here, it determines the likelihood of observing a Z statistic more significant than the calculated one, assuming the null hypothesis is accurate.
Ensuring appropriate application of the one-tailed test is essential, as an incorrect choice (opting for a one-tailed when a two-tailed is needed) can lead to misleading conclusions. Thus, clarity in hypothesis directionality is key to effective statistical testing.

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Most popular questions from this chapter

The article "Caffeine Knowledge, Attitudes, and Consumption in Adult Women" (J. of Nutrition Educ., 1992: 179-184) reports the following summary data on daily caffeine consumption for a sample of adult women: \(n=47\), \(\bar{x}=215 \mathrm{mg}, s=235 \mathrm{mg}\), and range \(=5-1176\). a. Does it appear plausible that the population distribution of daily caffeine consumption is normal? Is it necessary to assume a normal population distribution to test hypotheses about the value of the population mean consumption? Explain your reasoning. b. Suppose it had previously been believed that mean consumption was at most \(200 \mathrm{mg}\). Does the given data contradict this prior belief? Test the appropriate hypotheses at significance level . 10 and include a \(P\)-value in your analysis.

Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high-pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than \(.05 \mathrm{~mm}\). What hypotheses should be tested, and why? In this context, what are the type I and type II errors?

A sample of 12 radon detectors of a certain type was selected, and each was exposed to \(100 \mathrm{pCi} / \mathrm{L}\) of radon. The resulting readings were as follows: \(\begin{array}{rrrrrr}105.6 & 90.9 & 91.2 & 96.9 & 96.5 & 91.3 \\ 100.1 & 105.0 & 99.6 & 107.7 & 103.3 & 92.4\end{array}\) a. Does this data suggest that the population mean reading under these conditions differs from 100 ? State and test the appropriate hypotheses using \(\alpha=.05\). b. Suppose that prior to the experiment a value of \(\sigma=7.5\) had been assumed. How many determinations would then have been appropriate to obtain \(\beta=.10\) for the alternative \(\mu=95\) ?

For each of the following assertions, state whether it is a legitimate statistical hypothesis and why: a. \(H: \sigma>100\) b. \(H: \tilde{x}=45\) c. \(H: s \leq .20\) d. \(H: \sigma_{1} / \sigma_{2}<1\) e. \(H: \bar{X}-\bar{Y}=5\) f. \(H: \lambda \leq .01\), where \(\lambda\) is the parameter of an exponential distribution used to model component lifetime

Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40 -amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40 . If the mean amperage is lower than 40 , customers will complain because the fuses require replacement too often. If the mean amperage is higher than 40 , the manufacturer might be liable for damage to an electrical system due to fuse malfunction. To verify the amperage of the fuses, a sample of fuses is to be selected and inspected. If a hypothesis test were to be performed on the resulting data, what null and alternative hypotheses would be of interest to the manufacturer? Describe type I and type II errors in the context of this problem situation.
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