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Chapter 10: Problem 51

The true average diameter of ball bearings of a certain type is supposed to be \(0.5 \mathrm{in}\). What conclusion is appropriate when testing \(H_{0}: \mu=0.5\) versus \(H_{a}: \mu \neq 0.5\) in each of the following situations: a. \(n=13, t=1.6, \alpha=.05\) b. \(n=13, t=-1.6, \alpha=.05\) c. \(n=25, t=-2.6, \alpha=.01\) d. \(n=25, t=-3.6\)

Short Answer

Expert verified
Based on the statistical analysis and hypothesis testing for each situation, we can conclude: a) We cannot reject the null hypothesis that the average diameter of ball bearing is 0.5 in. b) We again cannot reject the null hypothesis that the average diameter of the ball bearing is 0.5 in. c) We do not reject the null hypothesis that the average diameter of ball bearing is 0.5in. d) We reject the null hypothesis that the average diameter of ball bearing is 0.5in.

Step by step solution

01

Understanding Hypothesis Testing

Hypothesis testing is a statistical method that is used in making statistical decisions using experimental data. In this exercise, we are testing two hypotheses. The null hypothesis \(H_{0}: \mu = 0.5\) suggests the average diameter of ball bearing is 0.5 in. The alternative hypothesis \(H_{a}: \mu \neq 0.5\) suggests that the average diameter is not 0.5 in.
02

Computing Critical Value for each situation

For hypothesis testing, we first calculate a 't' score and then find a critical value associated to our significance level 伪 which in our situation a and b is 0.05 and in c is 0.01. This critical value is then compared to our calculated 't' statistic. If absolute value of our calculated 't' statistic is greater than the critical value, we reject the null hypothesis. The degree of freedom for t-distribution is \(n-1\). For each situation, the critical values (two-tailed) for degrees of freedom 12 and 24 and significance level 0.05 and 0.01 are approximately 2.179, 2.064, and 2.797, respectively according to t-distribution table.
03

Comparing the test statistic with the Critical Value

Next, we compare the test statistic (t) with the critical value for each situation. a. The absolute value of the test statistic \(|1.6|\) is less than the critical value 2.179, so we cannot reject the null hypothesis. b. The absolute value of the test statistic \(|-1.6|\) is still less than the critical value 2.179, so we again cannot reject the null hypothesis. c. The absolute value of the test statistic \(|-2.6|\) is less than the critical value 2.797, so we do not reject the null hypothesis. d. There is no specific significance level mentioned in this case. However, if we assume the significance level as 0.05 (or even 0.01), the absolute value of the test statistic \(|-3.6|\) is more than the critical value of 2.064 (or 2.797), so we do reject the null hypothesis in this case.

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Key Concepts

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

Null Hypothesis
The Null Hypothesis, denoted by \( H_0 \), forms the foundation of hypothesis testing. In simple terms, it's the hypothesis that assumes no effect or no difference. In our exercise, the null hypothesis \( H_0: \mu = 0.5 \) posits that the true average diameter of the ball bearings is 0.5 inches. This simplicity is key. By starting with the assumption that there is no change or effect, we can objectively analyze whether the data provides enough evidence to suggest otherwise. Always keep in mind, the null hypothesis is what we aim to nullify or disprove.
Critical Value
Understanding the critical value is crucial in determining whether to accept or reject the null hypothesis. It acts as a benchmark in hypothesis testing. The critical value is derived from the significance level and the chosen distribution. If our computed test statistic (like the 't' statistic) is more extreme than the critical value, we reject the null hypothesis. For instance, in scenarios provided, the critical values were calculated from a t-distribution table based on degrees of freedom and significance level. In general, if the test statistic exceeds the critical value, it suggests there is a statistically significant effect.
Significance Level
The significance level, often represented by \( \alpha \), is a threshold we set to determine when to reject the null hypothesis. It's the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Typically, significance levels like 0.05 or 0.01 are used. For example, in our exercises, different \( \alpha \) levels such as 0.05 and 0.01 are chosen to decide the critical values. A lower significance level means we require stronger evidence against the null hypothesis to reject it. Always set the significance level before analyzing your data to avoid bias.
t-distribution
The t-distribution is a type of probability distribution that is symmetrical and bell-shaped, like the normal distribution, but it has thicker tails. It's especially useful when dealing with small sample sizes (usually less than 30) and is characterized by its degrees of freedom, \( n-1 \), where \( n \) is the number of observations. In this case, our t-distribution helps determine the critical values needed for hypothesis testing. The thicker tails mean it accounts more for variability, which is helpful in cases where the population standard deviation is unknown and the sample size is small.
Two-tailed Test
A two-tailed test is used when we want to determine if there is any significant difference in either direction, whether greater or less than a certain value. It assesses whether the sample parameter is either significantly greater than or less than the hypothesized population parameter. In the provided exercise, this is illustrated by \( H_a: \mu eq 0.5 \), meaning we're looking for any deviation from 0.5 inches, either upwards or downwards. This makes a two-tailed test more conservative since it divides the significance level by two, requiring evidence from either direction to potentially reject the null hypothesis.

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Most popular questions from this chapter

The amount of shaft wear after a fixed mileage was determined for each of 7 randomly selected internal combustion engines, resulting in a mean of \(0.0372\) in. and a standard deviation of \(0.0125 \mathrm{in}\). a. Assuming that the distribution of shaft wear is normal, test at level \(.05\) the hypotheses \(H_{0}: \mu=.035\) versus \(H_{a}:\) \(\mu>.035\). b. Using \(\sigma=0.0125, \alpha=.05\), and Appendix Table 5 , what is the approximate value of \(\beta\), the probability of a Type II error, when \(\mu=.04\) ? c. What is the approximate power of the test when \(\mu=\) \(.04\) and \(\alpha=.05 ?\)

Do state laws that allow private citizens to carry concealed weapons result in a reduced crime rate? The author of a study carried out by the Brookings Institution is reported as saying, "The strongest thing I could say is that \(\bar{I}\) don't see any strong evidence that they are reducing crime" (San Luis Obispo Tribune, January 23,2003 ).

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 \(p\) be the sample proportion of responses indicating no repair (so that no repair is identified with a success). Let \(\pi\) denote the true 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 \(\pi<.9 .\) The appropriate hypotheses are then \(H_{0}: \pi=.9\) versus \(H_{a}: \pi<.9\). 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.

Consider the following quote from the article "Review Finds No Link Between Vaccine and Autism" (San Luis Obispo Tribune, October 19,2005 ): " 'We found no evidence that giving MMR causes Crohn's disease and/or autism in the children that get the MMR,' said Tom Jefferson, one of the authors of The Cochrane Review. 'That does not mean it doesn't cause it. It means we could find no evidence of it." (MMR is a measles-mumps-rubella vaccine.) In the context of a hypothesis test with the null hypothesis being that MMR does not cause autism, explain why the author could not just conclude that the MMR vaccine does not cause autism.

According to a survey of 1000 adult Americans conducted by Opinion Research Corporation, 210 of those surveyed said playing the lottery would be the most practical way for them to accumulate \(\$ 200,000\) in net wealth in their lifetime ("One in Five Believe Path to Riches Is the Lottery," San Luis Obispo Tribune, January 11,2006 ). Although the article does not describe how the sample was selected, for purposes of this exercise, assume that the sample can be regarded as a random sample of adult Americans. Is there convincing evidence that more than \(20 \%\) of adult Americans believe that playing the lottery is the best strategy for accumulating \(\$ 200,000\) in net wealth?
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