Final answer:
The P-values for the provided z test statistics can be found using a standard normal distribution table or software; they represent the probability of observing a result as extreme as the test statistic under the null hypothesis. For z = -0.53, -0.98, -1.91, and -2.27 the corresponding P-values in a left-tailed test are 0.2981, 0.1635, 0.0281, and 0.0116 respectively, with the P-value for z = 1.30 not applicable for this left-tailed test.
Step-by-step explanation:
To find the P-value associated with each given z test statistic for the hypothesis test of the proportion of students at a university planning to purchase a meal plan, we look up the corresponding areas in the standard normal distribution table or use a statistical software. The null hypothesis H₀ states that p=0.20 and the alternative hypothesis Hₐ states that p<0.20. Since the test is left-tailed (Hₐ: p<0.20), the P-value for a given z statistic is the area to the left of the z value in the standard normal distribution.
- For z = -0.53, P-value ≈ 0.2981
- For z = -0.98, P-value ≈ 0.1635
- For z = -1.91, P-value ≈ 0.0281
- For z = -2.27, P-value ≈ 0.0116
- For z = 1.30, P-value is not applicable in this context as it implies an area to the right, and our test is for p<0.20, which corresponds to the left tail.
P-values are typically found by looking up the negative z-score in the z-table, as we seek the cumulative probability up to that z-score. The P-value is thus interpreted as the probability of getting a test statistic as extreme as, or more extreme than, the observed one, given that the null hypothesis is true. Whenever P-values are close to the significance level, it is recommended to calculate the exact P-value if possible.