Question

Newly purchased automobile tires of a certain type are supposed to be filled to a pressure of 30 psi. Let \(\mu\) denote the true average pressure. Find the \(P\) -value associated with each of the following given \(z\) statistic values for testing \(H_{0}: \mu=30\) versus \(H_{a}: \mu \neq 30\) when \(\sigma\) is known: a. \(2.10\) d. \(1.44\) b. \(-1.75\) e. \(-5.00\) c. \(0.58\)

Short answer

The corresponding P-values for the given Z-values are as follows: for Z=2.10 is 0.0358, for Z=1.44 is 0.1498, for Z=-1.75 is 1.9198, for Z=-5.00 is 2, and for Z=0.58 is 0.5620.

Step-by-step solution

Understanding the Terminology

The first step to solve this exercise is understanding the given values. Here, \(\mu\) is the true average pressure, \(\sigma\) is the standard deviation which is a known value. The aim is to test the null hypothesis \(H_{0}: \mu=30\) versus the alternative hypothesis \(H_{a}: \mu \neq 30\). For that, we are given z-statistic values and we need to find the corresponding P-values for each.

Calculating P-values using Z-table

The P-value can be calculated using a standard normal table (or Z-table), which gives the probabilities of various outcomes. For a two-tailed test, as it's in this case, the P-value is the probability that the z-score is greater than the absolute value of the given statistic, multiplied by 2. This is due to the symmetric property of the normal distribution.

Calculating P-value for Z=2.10

First, find the probability for Z=2.10 from the Z-table, which is 0.9821, then subtract it from 1 to get the probability which equals to 1-0.9821=0.0179. Since this is a two-tailed test, multiply this answer by 2. Therefore, P-value = 2*0.0179 = 0.0358 .

Calculating P-value for Z=1.44

From the Z-table, the probability for Z=1.44 is 0.9251. Subtract it from 1 to get the probability which equals to 1-0.9251=0.0749. Since it is a two-tailed test, multiply this answer by 2. Therefore, P-value = 2*0.0749 = 0.1498.

Calculating P-value for Z=-1.75

From the Z-table, the probability for Z=-1.75 is 0.0401. Subtract it from 1 to get the probability which equals to 1-0.0401=0.9599. Since it is a two-tailed test, multiply this answer by 2. Therefore, P-value = 2*0.9599 = 1.9198.

Calculating P-value for Z=-5.00

For Z=-5.00, the probability is very close to 0. However, subtract it from 1 to get the probability which equals to approximately 1. Since it is a two-tailed test, multiply this answer by 2. Therefore, P-value = 2*1 = 2.

Calculating P-value for Z=0.58

From the Z-table, the probability for Z=0.58 is 0.7190. Subtract it from 1 to get the probability which equals to 1-0.7190=0.2810. Since it is a two-tailed test, multiply this answer by 2. Therefore, P-value = 2*0.2810 = 0.5620.