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Chapter 8: Problem 166

Use a computer or calculator to select 36 random numbers from a normal distribution with mean 100 and standard deviation \(15 .\) Find the sample mean and \(z *\) for testing a two-tailed hypothesis test of \(\mu=100 .\) Using \(\alpha=0.05,\) state the decision. Repeat several times as in Table \(8.12 .\) Describe your findings.

Short Answer

Expert verified
Without specific random numbers, an explicit short answer can't be given. However, after several repetitions and analysis, a decision to accept or reject the null hypothesis can be made. If the absolute value of z-score exceeds the z-value at the given \(\alpha\) level, the null hypothesis that the population mean equals 100 will be rejected.

Step by step solution

01

Generate Random Samples

Use a random number generator with a normal distribution setting. Set the mean to 100 and the standard deviation to 15. Remember that these are the numbers that will be used to represent the population. Generate 36 numbers using these settings.
02

Calculate Sample Mean

Calculate the mean (average) of these 36 numbers. This will be your sample mean (\(\overline{X}\)).
03

Determine Z-Score

Next, determine the z-score using the formula \(z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}}\), where \(\overline{X}\) is the sample mean, \(\mu\) is the population mean (100 in this case), \(\sigma\) is the standard deviation (given as 15), and \(n\) is the sample size (36).
04

Conduct Hypothesis Test

Conduct a two-tailed hypothesis test using the calculated z-score and a significance level (\(\alpha\)) of 0.05. If the absolute value of the z-score is greater than the z-value corresponding to the \(\alpha\) level in the z-table (approximately 1.96 for a two-tailed test with \(\alpha = 0.05\)), reject the null hypothesis that the population mean is 100. Otherwise, fail to reject the null hypothesis.
05

Repeat and Describe Findings

Repeat the process several times, each time generating a new set of 36 random numbers and conducting the hypothesis test. Describe your findings. You should notice that the decision whether to reject the null hypothesis may vary from one sample to the next, due the randomness of the sampling process.

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Key Concepts

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

Normal Distribution
The normal distribution, also known as the Gaussian distribution, is a bell-shaped curve that is symmetrical about the mean. It is a fundamental concept in statistics and is used to describe how data is dispersed around a central value. In a perfectly normal distribution, the mean, median, and mode are all the same, located at the center of the curve.

Most values cluster around the mean, and as you move away, the frequency of the values decreases. Understanding the normal distribution is crucial for hypothesis testing because it allows statisticians to determine how likely a particular sample is to occur within a population. When data follows a normal distribution, we can use properties of the distribution, such as the empirical rule, to make inferences about probabilities and to perform hypothesis tests.
Sample Mean Calculation
The sample mean, denoted as \(\overline{X}\), is the average value of a sample drawn from a population. Calculating it is straightforward: sum all the data points in the sample and divide by the number of points. In the context of our exercise, where you have 36 random numbers, you add all those numbers together and then divide by 36. This represents the mean of your sample.

Importance of Sample Mean in Hypothesis Testing

  • The sample mean is used as an estimate of the population mean.
  • It's essential for comparing the sample to the population.
  • It's used to calculate the z-score and test statistics in hypothesis tests.
Proper calculation ensures accurate results in statistical analysis, making it a cornerstone of reliable hypothesis testing.
Z-Score
The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a z-score is 0, it indicates that the data point's score is identical to the mean score. A z-score can be positive or negative, indicating whether the value lies above or below the mean, respectively.

To calculate the z-score in hypothesis testing, you use the formula \(z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}}\), where \(\overline{X}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the standard deviation, and \(n\) is the sample size. This measurement is a crucial step in determining how unusual or typical a result is within the context of the hypothesis test.
Significance Level
The significance level, denoted by \(\alpha\), is the threshold used to judge whether a test statistic is sufficiently extreme to reject the null hypothesis. It's a measure of the degree of certainty in a hypothesis test; typically, a level of 0.05 is used, meaning there is a 5% risk of concluding that a difference exists when there is none (Type I error).

In hypothesis testing:
  • A low significance level means requiring stronger evidence to reject the null hypothesis.
  • If the calculated test statistic, such as a z-score, is more extreme than what the significance level allows, you reject the null hypothesis.
  • Conversely, if the statistic does not reach the significance level, you fail to reject the null hypothesis, meaning not enough evidence was found.
The choice of significance level affects the conclusion of the hypothesis test and should be determined before conducting the test.

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Most popular questions from this chapter

Use a computer and generate 50 random samples, each of size \(n=28,\) from a normal probability distribution with \(\mu=18\) and \(\sigma=4\) a. Calculate the \(z\) * corresponding to each sample mean. b. In regard to the \(p\) -value approach, find the proportion of \(50 z \star\) values that are "more extreme" than the \(z=-1.04\) that occurred in Exercise 8.201 \(\left(H_{a}: \mu \neq 18\right) .\) Explain what this proportion represents. c. In regard to the classical approach, find the critical values for a two- tailed test using \(\alpha=0.01 ;\) find the proportion of \(50 z \star\) values that fall in the critical region. Explain what this proportion represents.

a. A one-tailed hypothesis test is to be completed at the 0.05 level of significance. What calculated values of \(p\) will cause a rejection of \(H_{o} ?\) b. A two-tailed hypothesis test is to be completed at the 0.02 level of significance. What calculated values of \(p\) will cause a "fail to reject \(H_{o}\) " decision?

According to the Center on Budget and Policy Priorities' article "Curbing Flexible Spending Accounts Could Help Pay for Health Care Reform" (revised June \(10,2009),\) flexible-spending accounts encourage the over consumption of health care. People buy things they do not need; otherwise they lose the money. In 2007 , for those who did not use all of their account (about one out of every seven), the average amount lost was \(\$ 723\) Suppose a random sample of 150 employees who did not use all of their funds in 2009 is taken and an average amount of \(\$ 683\) was lost. Test the hypothesis that there is no significant difference in the average amount forfeited. Assume that \(\sigma=\$ 307\) per year. Use \(\alpha=0.05\) a. Define the parameter. b. State the null and alternative hypotheses. c. Specify the hypothesis test criteria. d. Present the sample evidence. e. Find the probability distribution information. f. Determine the results.

Use a computer and generate 50 random samples, each of size \(n=28,\) from a normal probability distribution with \(\mu=19\) and \(\sigma=4\) a. Calculate the \(z\) * corresponding to each sample mean that would result when testing the null hypothesis \(\mu=18\) b. In regard to the \(p\) -value approach, find the proportion of \(50 z \star\) values that are "more extreme" than the \(z=-1.04\) that occurred in Exercise \(8.201\left(H_{a}: \mu \neq 18\right)\) Explain what this proportion represents. c. In regard to the classical approach, find the critical values for a two- tailed test using \(\alpha=0.01 ;\) find the proportion of \(50 z \star\) values that fall in the noncritical region. Explain what this proportion represents.

Determine the critical region and the critical values used to test the following null hypotheses: a. \(\quad H_{o}: \mu=55(\geq), H_{a}: \mu<55, \alpha=0.02\) b. \(\quad H_{o}: \mu=-86(\geq), H_{a}: \mu<-86, \alpha=0.01\) c. \(\quad H_{o}: \mu=107, H_{a}: \mu \neq 107, \alpha=0.05\) d. \(\quad H_{o}: \mu=17.4(\leq), H_{a}: \mu>17.4, \alpha=0.10\)
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