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Chapter 10: Problem 4

Determine the critical value for a left-tailed test of a population mean with \(\sigma\) unknown at the \(\alpha=0.05\) level of significance with 19 degrees of freedom.

Short Answer

Expert verified
The critical value is approximately \( -1.729 \) for 19 degrees of freedom at \( \alpha = 0.05 \).

Step by step solution

01

Understand the Problem

You need to find the critical value for a left-tailed t-test of a population mean with an unknown population standard deviation. The significance level is \( \alpha=0.05 \) and there are 19 degrees of freedom.
02

Find the t-Distribution Table

Use a t-distribution table to find the critical value. The t-distribution is used instead of the z-distribution because the population standard deviation \( \sigma \) is unknown. The degrees of freedom (df) are 19.
03

Locate the Significance Level \( \alpha \)

In the t-distribution table, locate the column for \( \alpha = 0.05 \) in a one-tailed test. This column gives the critical values for a left-tailed test at the 0.05 level of significance.
04

Intersection with Degrees of Freedom

Find the row in the t-distribution table that corresponds to 19 degrees of freedom. Locate the intersection of this row with the column for \( \alpha = 0.05 \). This provides the critical t-value.
05

Confirm the Critical Value

The intersection point in the table shows the critical value. For 19 degrees of freedom and \( \alpha = 0.05 \), the critical value is approximately \( -1.729 \). It鈥檚 negative because it is a left-tailed test.

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Key Concepts

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

t-distribution
When we talk about the t-distribution, we're referring to a type of probability distribution that is symmetrical around the mean. Unlike the normal distribution, the t-distribution has heavier tails, which means it has a greater chance of producing values far from its mean. This makes it especially useful when dealing with small sample sizes where the population standard deviation (\( \sigma \)) is unknown.

The t-distribution is crucial in hypothesis testing, particularly for the t-test. It's used because it accounts for the variability introduced by estimating the population standard deviation from the sample. As sample size increases, the t-distribution approaches the shape of the normal distribution.

Key points about the t-distribution:
  • Symmetrical and bell-shaped, like the normal distribution
  • Heavier tails than the normal distribution, providing a higher probability for extreme values
  • Defined by degrees of freedom which influence its shape
  • Used primarily in small sample sizes when the population standard deviation is unknown.
degrees of freedom
Degrees of freedom (df) are a concept used in statistical analyses to describe the number of independent values that can vary in an analysis without breaking any constraints. In the context of the t-distribution, degrees of freedom are calculated as the sample size minus one ( ).

Degrees of freedom are crucial because they affect the shape of the t-distribution. With fewer degrees of freedom, the t-distribution is wider and has heavier tails. As the degrees of freedom increase, the t-distribution narrows and resembles a normal distribution more closely.

Practical points about degrees of freedom:
  • It essentially boils down to the number of datapoints that are free to vary
  • In smaller samples, fewer degrees of freedom result in a broader distribution
  • Understanding degrees of freedom helps in choosing the correct t-distribution for hypothesis testing
significance level
The significance level () is the probability of rejecting the null hypothesis when it is true. It is a threshold set by the researcher to determine when there is enough evidence to conclude that an effect or difference exists.

A common significance level is 0.05, which means there's a 5% risk of concluding that a difference exists when there actually is none. In a left-tailed test, this value marks the boundary below which we would reject the null hypothesis in favor of the alternative hypothesis.

Key aspects of the significance level:
  • It is denoted by alpha ( )
  • It defines the rejection region for the hypothesis test
  • Common values are 0.01, 0.05, and 0.10, with 0.05 being particularly standard
  • In a left-tailed test, we look at the lower tail; for a right-tailed or two-tailed test, it differs accordingly

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Most popular questions from this chapter

Carl Reinhold August Wunderlich said that the mean temperature of humans is \(98.6^{\circ} \mathrm{F}\). Researchers Philip Mackowiak, Steven Wasserman, \(\begin{array}{lllllll}\text { and } & \text { Myron } & \text { Levine } & {[\mathrm{JAMA},} & \text { Sept. } & 23-30 & 1992\end{array}\) \(268(12): 1578-80]\) thought that the mean temperature of humans is less than \(98.6^{\circ}\) F. They measured the temperature of 148 healthy adults 1 to 4 times daily for 3 days, obtaining 700 measurements. The sample data resulted in a sample mean of \(98.2^{\circ} \mathrm{F}\) and a sample standard deviation of \(0.7^{\circ} \mathrm{F}\) (a) Test whether the mean temperature of humans is less than \(98.6^{\circ} \mathrm{F}\) at the \(\alpha=0.01\) level of significance using the classical approach. (b) Determine and interpret the \(P\) -value.

Determine the critical value for a two-tailed test of a population mean with \(\sigma\) unknown at the \(\alpha=0.05\) level of significance with 12 degrees of freedom.

Lipitor The drug Lipitor is meant to reduce total cholesterol and LDL cholesterol. In clinical trials, 19 out of 863 patients taking \(10 \mathrm{mg}\) of Lipitor daily complained of flulike symptoms. Suppose that it is known that \(1.9 \%\) of patients taking competing drugs complain of flulike symptoms. Is there sufficient evidence to conclude that more than \(1.9 \%\) of Lipitor users experience flulike symptoms as a side effect at the \(\alpha=0.01\) level of signifi-cance?

Confidence in Schools In \(1995,40 \%\) of adults aged 18 years or older reported that they had "a great deal" of confidence in the public schools. On June \(1,2005,\) the Gallup Organization released results of a poll in which 372 of 1004 adults aged 18 years or older stated that they had "a great deal" of confidence in the public schools. Does the evidence suggest at the \(\alpha=0.05\) level of significance that the proportion of adults aged 18 years or older having "a great deal" of confidence in the public schools is significantly lower in 2005 than the 1995 proportion?

In Problems \(3-8,\) test the hypothesis, using (a) the classical approach and then (b) the P-value approach. Be sure to verify the requirements of the test. $$\begin{aligned}&H_{0}: p=0.3 \text { versus } H_{1}: p>0.3\\\&n=200 ; x=75 ; \alpha=0.05\end{aligned}$$
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