Scores on a common final exam in a large enrollment,...

Scores on a common final exam in a large enrollment, multiple-section freshman course are normally distributed with mean 72.7 and standard deviation $13.1 .$ a. Find the probability that the score $X$ on a randomly selected exam paper is between 70 and 80 . b. Find the probability that the mean score $x-$ of 38 randomly selected exam papers is between 70 and 80 .

Scores on a common final exam in a large enrollment, multiple-section freshman course are normally distributed with mean 72.7 and standard deviation $13.1 .$
a. Find the probability that the score $X$ on a randomly selected exam paper is between 70 and 80 .
b. Find the probability that the mean score $x-$ of 38 randomly selected exam papers is between 70 and 80 .

Key Concepts

Z-score

A z-score represents the number of standard deviations a data point is from the mean in a standard normal distribution. It is used to transform a normal distribution into the standard normal distribution, enabling calculation of probabilities by comparing the relative position of a value within the distribution.

Sampling Distribution of the Mean

When taking samples from a population, the distribution of the sample means forms a sampling distribution. This distribution has its mean equal to the population mean and its spread (standard error) smaller than the population standard deviation, becoming approximately normal even if the original distribution is not (Central Limit Theorem).

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve defined by the mean and standard deviation. It is widely used in statistics because many natural phenomena tend to be normally distributed, and it serves as a basis for several statistical tests and confidence interval calculations.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the population's distribution, the distribution of the sample mean will approximate a normal distribution as the sample size grows, provided the samples are independent and identically distributed. This is crucial for making inferences about population parameters from sample statistics.

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