Question

Newly purchased automobile tires of a certain type are supposed to be filled to a pressure of 30 psi. Let μ denote the true average pressure. Find the P-value associated with each of the following given z statistic values for testing H0: μ = 30 versus Ha: μ 30 when σ is known. (Give the answers to four decimal places.)

Answer 1

The P-value corresponding to a z statistic of -2.315 for a two-tailed test is 0.0103. This value indicates the probability of observing a sample mean at least as extreme as the test statistic, assuming the null hypothesis is true.

Step-by-step explanation:

The question requires us to find the P-value associated with a given z statistic value. In hypothesis testing, the P-value helps us determine if we can reject the null hypothesis, denoted as H0, in favor of the alternative hypothesis, Ha. Here, the null hypothesis is that the true average pressure of the automobile tires is 30 psi. The alternative hypothesis Ha: µ≠ 30 suggests the true average is not equal to 30 psi.

Given the z statistic (z = -2.315), we find the corresponding P-value by looking at the area under the normal distribution curve to the left of this z value for a two-tailed test since our Ha: µ≠ 30 suggests a non-directional alternative. A z score of -2.315 corresponds to a P-value of 0.0103. This means that if the true tire pressure were 30 psi, there is a probability of 0.0103 or 1.03% that we would observe a sample mean that differs from the population mean by at least as much as our test statistic, purely by chance.

Answer 2

Complete Question

Newly purchased automobile tires of a certain type are supposed to be filled to a pressure of 30 psi. Let μ denote the true average pressure. Find the P-value associated with each of the following given z statistic values for testing H0: μ = 30 versus Ha: μ 30 when σ is known. (Give the answers to four decimal places.)

calculate for each

(a) z = 2.30

(b) z = -1.7

Answer:

a

`p-value = 0.021448`

b

`p-value = 0.08913`

Step-by-step explanation:

From the question we are told that

The population mean is
`\mu = 30 \ psi`

The null hypothesis
`H_o : \mu = 30`

The alternative hypothesis is
`H_a : \mu \\e 30`

Considering question a

Here the test statistics is (a) z = 2.30

From the z table the probability of (Z > 2.30) is

`P(Z > 2.30 ) = 0.010724`

Generally the p-value is mathematically represented as

`p-value = 2 * P(Z > 2.30 )`

=>
`p-value = 2 * 0.010724`

=>
`p-value = 0.021448`

Considering question b

Here the test statistics is (a) z = -1.7

From the z table the probability of (Z < -1.7) is

`P(Z < -1.7 ) = 0.044565`

Generally the p-value is mathematically represented as

`p-value = 2 * P(Z < -1.70 )`

=>
`p-value = 2 * 0.044565`

=>
`p-value = 0.08913`