A toy manufactures is making toy balls for parties. The desired diameter for the

Question

A toy manufactures is making toy balls for parties. The desired diameter for the balls is 11.46 cm. Assume that the diameters of the balls can be considered to be normally distributed with population standard deviation of sigma = 0.0221 cm. A random sample of 20 balls was selected. It was found that the sample average diameter is 11.465 cm. The hypothesis H_0: mu = 11.46 cm should be tested against the alternative hypothesis H_1: mu notequalto 11.46 cm using a = 0.05. a) Determine the critical region for the test. Should the null hypothesis be rejected at the 5% significance level? Explain your conclusion. b) Use p-value to perform the test. Do you get the same conclusion? c) Construct a 95% confidence interval for mu. Does the confidence interval support your previous conclusion for the test?

Explanation / Answer

Null Hypothesis : H0 : = 11.46 cm

Alternative Hypothesis : Ha : 11.46 cm

(a) Critical region of the test will be calculated by test statistic. As the test is two - tailed here Z value will be equal to +-1.96

critical region will be outside of this region = xsample +- Z0.05 * (s/n) = 11.465 - 1.96 * (0.0221/20)

= 11.465 - 1.96 * 0.00494 = 11.455

so critical region for the test is <= 11.455 cm so as we see that null hypothesis value of population mean = 11.46 cm is not in this critical region, so we can not reject the null hypothesis.

(b) P - value =Pr ( <H ; xsample ; SE) =

Pr( <= 11.46 ; 11.465; 0.00494) = ?

Z = ( 11.46 - 11.465)/ 0.00494 = -1.012

so Pr ( <H ; xsample ; SE) = 0.1562

so P - value = 0.1562 which is not under significance level alpha <0.05 so we cannot reject the null Hypothesis.

(c) 95% confidence interval for = xsample +- Z0.05 (/n) = 11.465 +- 1.96 * ( 0.0221/20) =

(11.455, 11.475)

so this confidence interval contains the null hypothesis mean value so this support previous conclusion for the test

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