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 \\e 0.5025`

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

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 \\e 0.5025`

The test statistics is mathematically represented as

`t = (0.5046 - 0.5025)/( (0.0001)/(√(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

`(\alpha )/(2)` obtained from the normal distribution table is

`Z_{(\alpha )/(2) } = 1.96`

The margin of error is mathematically represented as

`E = Z_{(\alpha )/(2) }*(s)/(√(n) )`

=>
`E = 1.96 *(0.0001)/(√(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`