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Chapter 9: Problem 74

Explain when a sample is large enough to use the normal distribution to make a test of hypothesis about the population proportion.

Short Answer

Expert verified
A sample is large enough to use the normal distribution for hypothesis testing about the population proportion if both conditions are met: \(n \cdot p \geq 10 \) and \(n \cdot (1 - p) \geq 10\). This ensures that the normal distribution can be used as a close approximation for the population.

Step by step solution

01

Identifying the Criteria

The size of the sample becomes critical when dealing with population proportions. There are two criteria generally used to determine if the sample size is large enough: \n\n- \(n \cdot p \geq 10 \), where \(n\) is the sample size and \(p\) is the population proportion.- \(n \cdot (1 - p) \geq 10\), this condition must also be true.
02

Applying the Criteria

These conditions suggest that both the expected number of successes (positive instances) and failures (negative instances) must be greater than 10. When both these conditions are met in a binominal distribution, it is safe to use the normal distribution as an approximation for the hypothesis test.
03

Importance of Large Sample Size

It's important to understand why does a large sample size matter: larger samples tend to provide a more accurate reflection of the population, reducing sampling error and providing a better approximation to the normal distribution.

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Most popular questions from this chapter

The mean balance of all checking accounts at a bank on December 31,2011, was \(\$ 850 .\) A random sample of 55 checking accounts taken recently from this bank gave a mean balance of \(\$ 780\) with a standard deviation of \(\$ 230 .\) Using a \(1 \%\) significance level, can you conclude that the mean balance of such accounts has decreased during this period? Explain your conclusion in words. What if \(\alpha=.025\) ?

For each of the following examples of tests of hypothesis about \(\mu\), show the rejection and nonrejection regions on the \(t\) distribution curve. a. A two-tailed test with \(\alpha=.01\) and \(n=15\) b. A left-tailed test with \(\alpha=.005\) and \(n=25\) c. A right-tailed test with \(\alpha=.025\) and \(n=22\)

Write the null and alternative hypotheses for each of the following examples. Determine if each is a case of a two-tailed, a left-tailed, or a right-tailed test. a. To test if the mean amount of time spent per week watching sports on television by all adult men is different from \(9.5\) hours b. To test if the mean amount of money spent by all customers at a supermarket is less than \(\$ 105\) c. To test whether the mean starting salary of college graduates is higher than \(\$ 47,000\) per year d. To test if the mean waiting time at the drive-through window at a fast food restaurant during rush hour differs from 10 minutes e. To test if the mean hours spent per week on house chores by all housewives is less than 30

According to an estimate, the average age at first marriage for women in the United States was \(26.1\) years in 2010 (Time, March 21,2011 ). A recent sample of 60 women from New Jersey who got married for the first time this year showed that their average age at first marriage was \(27.2\) years with a standard deviation of \(3.5\) years. Using a \(2.5 \%\) significance level and the critical-value approach, can you conclude that the average age for women in New Jersey who got married for the first time this year is higher than \(26.1\) years? Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value range and \(\alpha=.025\) ?

Brooklyn Corporation manufactures DVDs. The machine that is used to make these DVDs is known. to produce not more than \(5 \%\) defective DVDs. The quality control inspector selects a sample of \(200 \mathrm{DVDs}\) each week and inspects them for being good or defective. Using the sample proportion, the quality con trol inspector tests the null hypothesis \(p \leq .05\) against the alternative hypothesis \(p>.05\), where \(p\) is th proportion of DVDs that are defective. She always uses a \(2.5 \%\) significance level. If the null hypothesi: is rejected, the production process is stopped to make any necessary adjustments. A recent sample of 200 DVDs contained 17 defective DVD: Using a \(2.5 \%\) significance level, would you conclude that the prod" should be stoppe to make necessary adjustments b. Perform the test of part a using a \(1 \%\) significance level. Is your decision different from the one in.par Comment on the results of parts a and \(b\)
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