Question

The diameters of aluminum alloy rods produced on an extrusion machine are known to have a standard deviation of 0.0001 in. A random sample of 25 rods has an average diameter of 0.5046 in.1)Test the hypothesis that mean rod diameter is0.5025 in. Assume a two-sided alternative and use α=0.05.2)Find the P-value for this test. 3)Construct a 95% two-sided confidence interval on the mean rod diameter.

Answer 1

Complete Question

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Answer:

The null hypothesis is rejected this means that $\mu \ne 0.5025$

The 95% confidence interval is $0.5045608 < \mu < 0.5046392$

Step-by-step explanation:

From the question we are told that

The standard deviation is s= 0.0001

The sample size is n = 25

The sample mean is $\= x = 0.5046 \ in$

The population mean is $\mu = 0.5025 \ in$

The null hypothesis is $H_o : \mu = 0.5025$

The alternative hypothesis is $H_a : \mu \ne 0.5025$

The test statistics is mathematically represented as

$t = \frac{0.5046 - 0.5025}{ \frac{0.0001}{\sqrt{25} } }$

$t = 105$

So the p-value from the z-table is mathematically represented as

$p-value = 2 * P( z > 105)$

$p-value = 0.000$

seeing that

$p-value < \alpha$ we reject the null hypothesis

The critical value of

$\frac{\alpha }{2}$ obtained from the normal distribution table is

$Z_{\frac{\alpha }{2} } = 1.96$

The margin of error is mathematically represented as

$E = Z_{\frac{\alpha }{2} }*\frac{s}{\sqrt{n} }$

=> $E = 1.96 *\frac{0.0001}{\sqrt{25} }$

=> $E =3.92 *10^{-5}$

The 95% confidence level is mathematically represented as

$\= x - E < \mu < \= x + E$

=> $0.5046 - 3.92 *10^{-5} < \mu < 0.5046 + 3.92 *10^{-5}$

=> $0.5045608 < \mu < 0.5046392$