Question

suppose that the words per sentence in a book are normally distributed with an unknown mean and standard deviation. a random sample of 17 sentences is taken and gives a sample mean of 24 words and a sample standard deviation of 3 words.

Answer 1

The sampling distribution of the sample mean is approximately normal, with a mean close to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

Step-by-step explanation:

The subject of this question is Mathematics, and the grade level is High School.

To answer the question regarding the sampling distribution of the sample mean, we can use the Central Limit Theorem. According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normal regardless of the shape of the population distribution, as long as the sample size is large enough.

In this case, since we are drawing 100 samples of size 40 from a population with a mean of 50 and a standard deviation of 4, we would expect the sampling distribution of the sample mean to be approximately normal, with a mean close to the population mean of 50 and a standard deviation equal to the population standard deviation divided by the square root of the sample size, which is 4/√40.

Answer 2

In this mathematics question, a sample from a normally distributed population of sentence lengths in a book presents the findings of the sample mean and standard deviation. Since the sample size is small, a t-distribution would be used to estimate the population mean and to conduct inferential statistics.

Step-by-step explanation:

Given that the words per sentence in a book are normally distributed with an unknown mean and standard deviation, a sample of 17 sentences with a sample mean of 24 words and a sample standard deviation of 3 words provides us with information about the book's sentence length variability.

The central limit theorem tells us that the sampling distribution of the sample mean will approach a normal distribution as the sample size gets larger. In this case, since the sample size is less than 30, we would ideally use the t-distribution when estimating the population mean from the sample mean. Furthermore, confidence intervals or hypothesis testing can be conducted to make inferences about the population mean using the sample data.