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Chapter 10: Problem 2

Find the critical value(s) of \(t\) that specify the rejection region in these situations: a. A two-tailed test with \(\alpha=.01\) and \(12 d f\) b. A right-tailed test with \(\alpha=.05\) and \(16 d f\) c. A two-tailed test with \(\alpha=.05\) and \(25 d f\) d. A left-tailed test with \(\alpha=.01\) and \(7 d f\)

Short Answer

Expert verified
#Short Answer# a. The critical values for a two-tailed test with 伪 = 0.01 and 12 d.f. are approximately 卤3.106. b. The critical value for a right-tailed test with 伪 = 0.05 and 16 d.f. is approximately 1.746. c. The critical values for a two-tailed test with 伪 = 0.05 and 25 d.f. are approximately 卤2.060. d. The critical value for a left-tailed test with 伪 = 0.01 and 7 d.f. is approximately -2.998.

Step by step solution

01

Test Type

This is a two-tailed test. We will use 伪/2 = 0.005 to find the critical values on each tail. ##Step 2: Identify the degrees of freedom##
02

Degrees of Freedom

In this case, degrees of freedom (d.f.) = 12. ##Step 3: Find the critical value##
03

Critical Value

Using a t-distribution table or calculator, we find the critical values for 伪/2 = 0.005 and 12 d.f. The critical values are approximately 卤3.106. ##Step 4: Conclusion##
04

Conclusion

The critical values for a two-tailed test with 伪 = 0.01 and 12 d.f. are approximately 卤3.106. #b. A right-tailed test with 伪 = .05 and 16 df# ##Step 1: Determine the test type##
05

Test Type

This is a right-tailed test. We will use 伪 = 0.05. ##Step 2: Identify the degrees of freedom##
06

Degrees of Freedom

In this case, degrees of freedom (d.f.) = 16. ##Step 3: Find the critical value##
07

Critical Value

Using a t-distribution table or calculator, we find the critical value for 伪 = 0.05 and 16 d.f. The critical value is approximately 1.746. ##Step 4: Conclusion##
08

Conclusion

The critical value for a right-tailed test with 伪 = 0.05 and 16 d.f. is approximately 1.746. #c. A two-tailed test with 伪 = .05 and 25 df# ##Step 1: Determine the test type and 伪/2##
09

Test Type

This is a two-tailed test. We will use 伪/2 = 0.025. ##Step 2: Identify the degrees of freedom##
10

Degrees of Freedom

In this case, degrees of freedom (d.f.) = 25. ##Step 3: Find the critical value##
11

Critical Value

Using a t-distribution table or calculator, we find the critical values for 伪/2 = 0.025 and 25 d.f. The critical values are approximately 卤2.060. ##Step 4: Conclusion##
12

Conclusion

The critical values for a two-tailed test with 伪 = 0.05 and 25 d.f. are approximately 卤2.060. #d. A left-tailed test with 伪 = .01 and 7 df# ##Step 1: Determine the test type##
13

Test Type

This is a left-tailed test. We will use 伪 = 0.01. ##Step 2: Identify the degrees of freedom##
14

Degrees of Freedom

In this case, degrees of freedom (d.f.) = 7. ##Step 3: Find the critical value##
15

Critical Value

Using a t-distribution table or calculator, we find the critical value for 伪 = 0.01 and 7 d.f. The critical value is approximately -2.998. ##Step 4: Conclusion##
16

Conclusion

The critical value for a left-tailed test with 伪 = 0.01 and 7 d.f. is approximately -2.998.

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Key Concepts

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

Critical Value
The critical value in a t-distribution is an essential component in statistical hypothesis testing. It helps to determine whether the null hypothesis should be rejected. The critical value marks the boundaries of the rejection region, which is the area in the tails of the distribution that lies beyond a certain chance of error, known as the level of significance (\( \alpha \)).
  • For a two-tailed test, you need to divide the level of significance by two since the rejection area is split between both tails of the distribution.
  • For one-tailed tests, the entire significance level is in one tail.
  • Finding a critical value requires using a t-distribution table, or a calculator, based on degrees of freedom and the chosen significance level.
These critical values help decide whether the test statistic falls within the critical region, leading to the rejection of the null hypothesis.
Degrees of Freedom
Degrees of freedom (\( \text{d.f.} \)) are a crucial aspect of t-distributions and greatly impact the shape of the distribution. It essentially refers to the number of independent observations in a sample minus the number of parameters estimated. The calculation of degrees of freedom in most t-tests is straightforward:
  • For a single sample, the degrees of freedom are equal to the sample size minus one (\( n - 1 \)).
  • More complex designs, such as paired samples or multiple groups, may involve more intricate calculations.
Degrees of freedom affect the spread and the tail areas of the t-distribution. Lower degrees of freedom result in fatter tails, indicating more variability. As the degrees of freedom increase, the t-distribution approaches a normal distribution.
Two-Tailed Test
A two-tailed test is a statistical test where the area of rejection is on both ends of the sampling distribution. This type of test is used when we are interested in determining whether there is a difference regardless of direction.
  • In hypothesis testing, it looks for deviations in either direction from the hypothesized parameter.
  • For example, if testing whether a sample mean is different from a known population mean, you would check for values significantly higher or lower.
The significance level for a two-tailed test is divided between the two tails. So if \( \alpha = 0.05 \), then each tail would have a 0.025 level of significance. This approach is more conservative, as it requires stronger evidence to reject the null hypothesis since it checks both directions.
Right-Tailed Test
In a right-tailed test, the rejection region is in the right tail of the distribution. This type of test is specifically used when predicting an increase or greater values in the parameter of interest.
  • For example, if testing whether a new drug increases response times, a right-tailed test could be appropriate.
  • The critical value separates the non-rejection area from the rejection region in the right tail.
The entire chosen significance level (\( \alpha \)) is in the right side tail. Thus, for \( \alpha = 0.05 \), 5% of the distribution area falls under the rejection region. Right-tailed tests are pivotal in hypotheses where you only care about deviations in one direction, specifically to the right.
Left-Tailed Test
A left-tailed test, meanwhile, sets its sights on the left end of the distribution. It's most applicable when a decrease or lesser value in the parameter in question is the point of interest.
  • For example, if assessing whether a new teaching technique lowers anxiety levels, a left-tailed test could be used.
  • The critical value defines the boundary beyond which the null hypothesis will be rejected.
Here, the entire significance level (\( \alpha \)) resides in the left tail. For instance, with a significance level of 0.01, 1% of the area on the left tail makes up the rejection region. The left-tailed test is perfect for scenarios where changes suspected are directionally specific to the left.

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Most popular questions from this chapter

An experiment was conducted to compare the mean reaction times to twotypes of traffic signs: prohibitive (No Left Turn) and permissive (Left Turn Only). Ten drivers were included in the experiment. Each driver was presented with 40 traffic signs, 20 prohibitive and 20 permissive, in random order. The mean time to reaction (in milliseconds) was recorded for each driver and is shown here a. Explain why this is a paired-difference experiment and give reasons why the pairing should be useful in increasing information on the difference between the mean reaction times to prohibitive and permissive traffic signs. b. Use the Excel printout to determine whether there is a significant difference in mean reaction times to prohibitive and permissive traffic signs. Use the \(p\) -value approach.

An article in American Demographics investigated consumer habits at the mall. We tend to spend the most money shopping on the weekends, and, in particular, on Sundays from 4 to 6 P.M. Wednesday morning shoppers spend the least! \({ }^{19}\) Suppose that a random sample of 20 weekend shoppers and a random sample of 20 weekday shoppers were selected, and the amount spent per trip to the mall was recorded. $$ \begin{array}{lcc} & \text { Weekends } & \text { Weekdays } \\ \hline \text { Sample Size } & 20 & 20 \\ \text { Sample Mean (\$) } & 78 & 67 \\ \text { Sample Standard Deviation (\$) } & 22 & 20 \end{array} $$ a. Is it reasonable to assume that the two population variances are equal? Use the \(F\) -test to test this hypothesis with \(\alpha=.05\). b. Based on the results of part a, use the appropriate test to determine whether there is a difference in the average amount spent per trip on weekends versus weekdays. Use \(\alpha=.05\).

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A pharmaceutical manufacturer purchases a particular material from two different suppliers. The mean level of impurities in the raw material is approximately the same for both suppliers, but the manufacturer is concerned about the variability of the impurities from shipment to shipment. To compare the variation in percentage impurities for the two suppliers, the manufacturer selects 10 shipments from each of the two suppliers and measures the percentage of impurities in the raw material for each shipment. The sample means and variances are shown in the table. $$ \begin{aligned} &\begin{array}{ll} \text { Supplier } \mathrm{A} & \text { Supplier } \mathrm{B} \\ \hline \bar{x}_{1}=1.89 & \bar{x}_{2}=1.85 \\ s_{1}^{2}=.273 & s_{2}^{2}=.094 \end{array}\\\ &n_{1}=10 \quad n_{2}=10 \end{aligned} $$ a. Do the data provide sufficient evidence to indicate a difference in the variability of the shipment impurity levels for the two suppliers? Test using \(\alpha=.01 .\) Based on the results of your test, what recommendation would you make to the pharmaceutical manufacturer? b. Find a \(99 \%\) confidence interval for \(\sigma_{2}^{2}\) and interpret your results.

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