Performing a Sign Test In Exercises 7–22, (a) identify the cla...

Question

Performing a Sign Test In Exercises 7–22, (a) identify the claim and state Ho and Ha, (b) find the critical value, (c) find the test statistic, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

Hourly Wages A labor organization claims that the median hourly wage of podiatrists is at least $70.00. In a random sample of 23 podiatrists, 17 earn less than $70.00 per hour, 5 earn more than $70.00 per hour, and 1 earns $70.00 per hour. At , can you reject the labor organization’s claim? (Adapted from U.S. Bureau of Labor Statistics)

Accepted Answer

All right, hello, everyone. So this question says, a community center director claims that the median number of hours volunteers contribute per week is at least 10 hours. In a random sample of 25 volunteers, the number of hours they contributed last week, relative to 10 hours, is as follows. Part one says to identify the claim and stateh not an H sub a. And part two says, at the 0.05 significance level, can you reject the director's claim? Interpret the decision. So Here Right, the claim that's being stated, to answer part one of the question, is that the population median volunteer hours represented here by Tao is at least 10 hours, so greater than or equal to 10. So the null hypothesis H0 would state that tau is equal to 10. Whereas H A, the alternative hypothesis would state that tau. Is less than 10. And now to conduct the sign test at alpha equals 0.05. So, the first step in this process is to remove any ties. In other words, remove the number of volunteers that contributed exactly 10 hours. In this case, there were 2 of those. So the effective sample size N is equal to the original 25 subtracted by the two that were removed, which gives you 23. From here Relative to 10, any values that are greater would get positive signs, whereas any values that are smaller would get negative signs. Now, let's say that X is the number of positive signs. And in this case, there are 8 of those. There are 8 numbers or 8 volunteers rather that contributed more than 10 hours. Under the null hypothesis H knot, the number of positive signs follows a binomial distribution. Such that N is equal to 23. And P is equal to 0.5. Now, we're conducting a left-tailed test in this case, which means that we're looking for the probability of capital X being less than or equal to 8. Using either a binomial calculator or software, this probability is approximately equal to 0.1056. Which now leads you to make a decision. Our significance level is equal to 0.05, and our calculated P value is approximately 0.1056. Now Our probability, our P value 0.1056, is in fact greater than alpha, 0.05. Oops. And because it's greater than alpha, ultimately we fail to reject. not we fail to reject the null hypothesis. So now this leads you to the final answer, which reads as follows. For part one, H is tau is equal to 10, and H of A is tau is less than 10. For part two, failed to reject Hno. At the 5% significance level, there is insufficient evidence to conclude that the true median volunteer hours is less than 10%. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Hypothesis Testing

Sign Test

Critical Value and Test Statistic

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