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When conducting a one-sample z test, understanding the concept of population proportion is crucial. Population proportion refers to the ratio of individuals in a population who have a particular characteristic. In the context of our example, we're interested in the proportion of teens who rarely or never argue with their friends. Based on a larger survey, we know that this proportion is 72%. It's important for us to assume that this national proportion is generally reflective of other similar groups, like the large high school, provided that the sample selection and size conditions are appropriately met. This helps ensure the sample proportion can be a reliable estimate of the population proportion, allowing researchers to make inferences about the population from which the sample was drawn.

Random sampling is a key principle in statistics, essential for obtaining reliable and unbiased results. When we talk about random sampling, we mean that every individual in the population has an equal chance of being selected. In our exercise, a random sample of 150 students was chosen from a large high school. This random selection is critical because it helps to approximate the true diversity and characteristics of the entire school population. Random sampling minimizes bias, making the sample representative of the population, which is essential for the results of our one-sample z test to be valid. Therefore, confirming that a random sample was used allows us to proceed with confidence in our statistical analysis.

For a one-sample z test to be valid, the sample size needs to meet specific conditions. These ensure that the test results are reliable and the normal approximation is appropriate. In general, the sample size should be large enough so that both the expected number of successes (\( np \)) and the expected number of failures (\( n(1-p) \)) are at least 10. In our example, we have a sample size (\( n \)) of 150, with a national success proportion (\( p \)) of 0.72.Calculating gives us \( np = 150 \times 0.72 = 108 \) and \( n(1-p) = 150 \times 0.28 = 42 \), both values are comfortably above 10. This indicates our sample size is sufficient, supporting the validity of using the normal approximation for the z test.

Normal approximation involves using a normal distribution to approximate the sampling distribution of a sample statistic. In the context of a one-sample z test for a population proportion, this means treating the sample proportion as if it follows a normal distribution. For normal approximation to be valid, certain conditions, such as random sampling and appropriate sample size, must be met. With the sample size condition verified in the previous section, we can utilize the normal approximation for our z test. This approximation simplifies calculations and allows for easy interpretation of results, making it easier to test hypotheses about population proportions. When conditions are satisfied, normal approximation is a powerful tool for making statistical inferences.