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Chapter 8: Q20BB (page 356)

Finding Critical Values of \({\chi ^2}\) Repeat Exercise 19 using this approximation (with k and z as described in Exercise 19):

\({\chi ^2} = k{\left( {1 - \frac{2}{{9k}} + z\sqrt {\frac{2}{{9k}}} } \right)^3}\)

Short Answer

Expert verified

The estimated critical value is equal to 82.360. The value obtained is approximately equal to the value obtained using technology (82.292).

Step by step solution

01

Given information

A sample of the number of words spoken in a day is considered.

The sample size is equal to 56. The value of the degrees of freedom is equal to 55.

The value of the z-score is equal to 2.33. The actual critical value of \({\chi ^2}\) is equal to 82.292.

02

Compute the approximate critical value

The approximate critical value has the following formula:

\({\chi ^2} = k{\left( {1 - \frac{2}{{9k}} + z\sqrt {\frac{2}{{9k}}} } \right)^3}\).

The values are given as follows.

  • k is equal to 55.

Substitute the above values in the formula to obtain the critical value, as shown.
\(\begin{array}{c}{\chi ^2} = 55{\left( {1 - \frac{2}{{9 \times 55}} + 2.33 \times \sqrt {\frac{2}{{9 \times 55}}} } \right)^3}\\ \approx 82.360\end{array}\).

Thus, the critical value is equal to 82.360.

03

Comparison

The critical value of\({\chi ^2}\)obtained using the formula (82.360) isapproximately equalto the critical value of \({\chi ^2}\) obtained using technology (82.292).

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Most popular questions from this chapter

Using Technology. In Exercises 5鈥�8, identify the indicated values or interpret the given display. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section. Use = 0.05 significance level and answer the following:

a. Is the test two-tailed, left-tailed, or right-tailed?

b. What is the test statistic?

c. What is the P-value?

d. What is the null hypothesis, and what do you conclude about it?

e. What is the final conclusion?

Biometric Security In a USA Today survey of 510 people, 53% said that we should replace passwords with biometric security, such as fingerprints. The accompanying Statdisk display results from a test of the claim that half of us say that we should replace passwords with biometric security.

P-Values. In Exercises 17鈥�20, do the following:

a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.

b. Find the P-value. (See Figure 8-3 on page 364.)

c. Using a significance level of 伪 = 0.05, should we reject H0or should we fail to reject H0?

The test statistic of z = 2.01 is obtained when testing the claim that p鈮�0.345.

t Test Exercise 2 refers to a t test. What is the t test? Why is the letter t used? What is unrealistic about the z test methods in Part 2 of this section?

In Exercises 9鈥�12, refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favours heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favours heads (because it is easy to get 11 heads in 20 flips by chance with a fair coin).

Exercise 8 鈥淧ulse Rates鈥�

Testing Claims About Proportions. In Exercises 9鈥�32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Postponing Death An interesting and popular hypothesis is that individuals can temporarily postpone death to survive a major holiday or important event such as a birthday. In a study, it was found that there were 6062 deaths in the week before Thanksgiving, and 5938 deaths the week after Thanksgiving (based on data from 鈥淗olidays, Birthdays, and Postponement of Cancer Death,鈥� by Young and Hade, Journal of the American Medical Association, Vol. 292, No. 24). If people can postpone death until after Thanksgiving, then the proportion of deaths in the week before should be less than 0.5. Use a 0.05 significance level to test the claim that the proportion of deaths in the week before Thanksgiving is less than 0.5. Based on the result, does there appear to be any indication that people can temporarily postpone death to survive the Thanksgiving holiday?
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