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# Confidence Interval Estimate, Hypotheses and Test Statistics

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1. Mimi was the 5th seed in 2012 UMUC Tennis Open that took place in August. In this tournament, she won 83 of her 100 serving games. Based on UMUC Sports Network, she wins 80% of the serving games in her 5-year tennis career.

(a) Find a 95% confidence interval estimate of the proportion of serving games Mimi won. (Show work)

(b) Based on the confidence interval estimate you got in part (a), is this tournament result consistent with her career record of 80%? Why or why not? Please explain your conclusion.

(c) A sport reporter commented that Mimi's performance in the tournament is better than usual. You decide to test if the reporter's claim is valid by using hypothesis testing that you just learned from STAT 200 class. What are your null and alternative hypotheses?

(d) What is the test statistic? (Show work)

(e) What is the P-value? (Show work)

(f) What is the critical value? (Show work)

(g) What is your conclusion of the testing at 0.05 significance level? Why?

2. A simple random sample of 120 SAT scores has a mean of 1540. Assume that SAT scores have a population standard deviation of 333.

(a) Construct a 95% confidence interval estimate of the mean SAT score. (Show work)

(b) Is a 99% confidence interval estimate of the mean SAT score wider than the 95% confidence interval estimate you got from part (a)? Why? [You don't have to construct the 99% confidence interval]

3. The playing times of songs are normally distributed. Listed below are the playing times (in seconds) of 10 songs from a random sample. Use a 0.05 significance level to test the claim that the songs are from a population with a standard deviation less than 1 minute.

448 231 246 246 227 213 239 258 255 257

(a) What are your null hypothesis and alternative hypothesis?

(b) What is the test statistic? (Show work)

(c) What is your conclusion? Why? (Show work)

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#### Solution Preview

1. Mimi was the 5th seed in 2012 UMUC Tennis Open that took place in August. In this tournament, she won 83 of her 100 serving games. Based on UMUC Sports Network, she wins 80% of the serving games in her 5-year tennis career.
(a) Find a 95% confidence interval estimate of the proportion of serving games Mimi won. (Show work)

The mean proportion: 83/100=0.83
The critical value for 95% confidence interval is 1.96
Margin of error=1.96*sqrt(0.83*(1-0.83)/100)= 0.074.
The upper limit: 0.83+0.074=0.904
The lower limit: 0.83-0.074=0.756
Therefore, the 95% confidence interval for the proportion is [0.756, 0.904].

(b) Based on the confidence interval estimate you got in part (a), is this tournament result consistent with her career record of 80%? Why or why not? Please explain your conclusion.

The result is consistent with her career record of 0.80 since 0.80 is included in the confidence interval.
(c) A sport reporter commented that Mimi's performance in the tournament ...

#### Solution Summary

The expert examines confidence interval estimates, hypothesis and test statistics.

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