In a survey, 39% of the respondents stated that they talk to their pets on the telephone. A veterinarian believed this result to be too high, so she randomly selected 170 pet owners and discovered that 61 of them
spoke to their pet on the telephone. Does the veterinarian have a right to be skeptical? Use the a = 0.05 level of significance.
Find the p-value

Step 1 ：First, we need to calculate the sample proportion. In this case, 61 out of 170 pet owners talk to their pets on the telephone. So, the sample proportion is \(\frac{61}{170} = 0.3588\).

Step 2 ：Next, we need to calculate the standard error. The formula for the standard error of a proportion is \(\sqrt{\frac{p(1-p)}{n}}\), where p is the population proportion and n is the sample size. In this case, p is 0.39 (the proportion of pet owners who talk to their pets on the telephone according to the survey) and n is 170 (the number of pet owners in the veterinarian's sample). So, the standard error is \(\sqrt{\frac{0.39(1-0.39)}{170}} = 0.0363\).

Step 3 ：Then, we need to calculate the z-score. The formula for the z-score is \(\frac{\hat{p} - p}{SE}\), where \(\hat{p}\) is the sample proportion, p is the population proportion, and SE is the standard error. In this case, \(\hat{p}\) is 0.3588, p is 0.39, and SE is 0.0363. So, the z-score is \(\frac{0.3588 - 0.39}{0.0363} = -0.8595\).

Step 4 ：Finally, we need to find the p-value. The p-value is the probability of getting a z-score as extreme as the one we calculated, assuming the null hypothesis is true. In this case, the null hypothesis is that the population proportion is 0.39. We can find the p-value using a z-table or a statistical calculator. The p-value for a z-score of -0.8595 is 0.1951.

Step 5 ：Since the p-value (0.1951) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means that the veterinarian does not have enough evidence to be skeptical about the survey result. Therefore, the final answer is \(\boxed{0.1951}\).