The shape of the sampling distribution of (y - TN), representing the difference in the mean number of errors between the samples of people who took a driving class and those who did not, can be described as approximately normal. This is due to the application of the Central Limit Theorem, independence of sampling, and random sampling.
The shape of the sampling distribution of (y - TN), which represents the difference in the mean number of errors between the samples of people who took a driving class and those who did not, can be described as approximately normal. This is because the sampling distribution of the difference in means tends to follow a normal distribution under certain conditions, such as when the sample sizes are large enough.
Here's why we can expect the sampling distribution to be approximately normal:
1. Central Limit Theorem: According to the Central Limit Theorem, the sampling distribution tends to approximate a normal distribution regardless of the shape of the population distribution when the sample size is sufficiently large. In this case, the sample sizes of both groups (60 and 40) are reasonably large, so we can assume that the Central Limit Theorem applies.
2. Independence: It is assumed that the individuals in the sample are selected independently from each other. This means that the response or performance of one individual does not affect the response or performance of another. If the sampling process is conducted in a way that ensures independence, it contributes to the normality of the sampling distribution.
3. Random Sampling: The sampling is described as stratified random sampling, which means that individuals are selected randomly from each group. Random sampling helps to ensure that the sample is representative of the population and reduces the potential for bias.
When the conditions of the Central Limit Theorem, independence, and random sampling are met, the sampling distribution of (y - TN) will be approximately normal. This assumption of normality is important for making inferences and performing statistical tests on the difference in means between the two groups.