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Chapter 10: Problem 1

Explain why the statement \(\bar{x}=50\) is not a legitimate hypothesis.

Short Answer

Expert verified
The statement \(\bar{x} = 50\) is not a legitimate hypothesis because hypotheses are made about population parameters (like \(\mu\), \(\sigma^2\)), not sample statistics such as \(\bar{x}\). A correct version of the hypothesis would be \(H_0: \mu = 50\).

Step by step solution

01

Understanding the concepts of hypothesis

A hypothesis is a statement about a population parameter, not a sample statistic. For example, for a population mean, we use \(\mu\) to denote it in hypotheses, like \(H_0: \mu = \text{some value}\), \(H_1: \mu \neq \text{some value}\). So, the hypothesis has to be about population parameters, not the sample statistics.
02

Exploring the illegitimate hypothesis

The statement \(\bar{x} = 50\) is indicating a sample mean, not a population parameter. Statistically, we collect such sample data from our population to infer about the unknown population constants, such as population mean \(\mu\), population variance \(\sigma^2\), and so on. The hypothesis should be about these population constants, not the sample statistics.
03

Explaining the correct hypothesis

If we want to formulate a correct hypothesis for a population mean, and we suspect that the mean could be 50, instead of saying \(\bar{x} = 50\), we should have a hypothesis like \(H_0: \mu = 50\) or \(H_1: \mu \neq 50\) depending upon our initial claim and the kind of test (two-tailed, left-tailed, or right-tailed) we wish to perform.

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Most popular questions from this chapter

For which of the following \(P\) -values will the null hypothesis be rejected when performing a test with a significance level of .05: a. \(.001\) d. \(.047\) b. \(.021\) e. \(.148\) c. \(.078\)

A television manufacturer claims that (at least) \(90 \%\) of its TV sets will need no service during the first 3 years of operation. A consumer agency wishes to check this claim, so it obtains a random sample of \(n=100\) purchasers and asks each whether the set purchased needed repair during the first 3 years after purchase. Let \(\hat{p}\) be the sample proportion of responses indicating no repair (so that no repair is identified with a success). Let \(p\) denote the actual proportion of successes for all sets made by this manufacturer. The agency does not want to claim false advertising unless sample evidence strongly suggests that \(p<.9\). The appropriate hypotheses are then \(H_{0}: p=.9\) versus \(H_{a}: p<.9\). a. In the context of this problem, describe Type I and Type II errors, and discuss the possible consequences of each. b. Would you recommend a test procedure that uses \(\alpha=.10\) or one that uses \(\alpha=.01 ?\) Explain.

A hot tub manufacturer advertises that with its heating equipment, a temperature of \(100^{\circ} \mathrm{F}\) can be achieved on average in 15 minutes or less. A random sample of 25 tubs is selected, and the time necessary to achieve a \(100^{\circ} \mathrm{F}\) temperature is determined for each tub. The sample mean time and sample standard deviation are \(17.5\) minutes and \(2.2\) minutes, respectively. Does this information cast doubt on the company's claim? Carry out a test of hypotheses using significance level \(.05\).

Students at the Akademia Podlaka conducted an experiment to determine whether the Belgium-minted Euro coin was equally likely to land heads up or tails up. Coins were spun on a smooth surface, and in 250 spins, 140 landed with the heads side up (New Scientist, January 4,2002 ). Should the students interpret this result as convincing evidence that the proportion of the time the coin would land heads up is not. 5? Test the relevant hypotheses using \(\alpha=.01\). Would your conclusion be different if a significance level of \(.05\) had been used? Explain.

Give as much information as you can about the \(P\) -value of a \(t\) test in each of the following situations: a. Two-tailed test, \(\mathrm{df}=9, t=0.73\) b. Upper-tailed test, \(\mathrm{df}=10, t=-0.5\) c. Lower-tailed test, \(n=20, t=-2.1\) d. Lower-tailed test, \(n=20, t=-5.1\) e. Two-tailed test, \(n=40, t=1.7\)
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