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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

In your own words, explain the difference between "beyond all reasonable doubt" and "beyond all doubt."

To test \(H_{0}: \mu=4.5\) versus \(H_{1}: \mu>4.5,\) a random sample of size \(n=13\) is obtained from a population that is known to be normally distributed with \(\sigma=1.2\) (a) If the sample mean is determined to be \(\bar{x}=4.9,\) compute and interpret the \(P\) -value. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.1\) level of significance, will the researcher reject the null hypothesis? Why?

The following is a quotation from Sir Ronald A. Fisher, a famous statistician. "For the logical fallacy of believing that a hypothesis has been proved true, merely because it is not contradicted by the available facts, has no more right to insinuate itself in statistics than in other kinds of scientific reasoning \(\ldots . .\) It would, therefore, add greatly to the clarity with which the tests of significance are regarded if it were generally understood that tests of significance, when used accurately, are capable of rejecting or invalidating hypotheses, in so far as they are contradicted by the data: but that they are never capable of establishing them as certainly true \(\ldots .\) " In your own words, explain what this quotation means.

To test \(H_{0}: \mu=45\) versus \(H_{1}: \mu \neq 45,\) a random sample of size \(n=40\) is obtained from a population whose standard deviation is known to be \(\sigma=8\) (a) Does the population need to be normally distributed to compute the \(P\) -value? (b) If the sample mean is determined to be \(\bar{x}=48.3\) compute and interpret the \(P\) -value. (c) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, will the researcher reject the null hypothesis? Why?

Farm Size In \(1990,\) the average farm size in Kansas was 694 acres, according to data obtained from the U.S. Department of Agriculture. A researcher claims that farm sizes are larger now due to consolidation of farms. She obtains a random sample of 40 farms and determines the mean size to be 731 acres. Assume that \(\sigma=212\) acres. Test the researcher's claim at the \(\alpha=0.05\) level of significance.
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