In the second quarter of 2015, the average movie ticket cost 8.61 . In a random sample of 50 mo

In the second quarter of 2015, the average movie ticket cost...

Key Concepts

Z-Score Calculation

The z-score represents the number of standard errors that a sample mean is away from the population mean. It is computed by subtracting the population mean from the sample mean and then dividing the result by the standard error. This standardized value facilitates the use of the standard normal distribution to determine probabilities associated with the sample mean.

Standard Error

The standard error is a measure of the dispersion or variability of the sample mean around the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size, offering a quantifiable metric to understand the precision of the sample mean as an estimate of the population mean.

Probability Calculations Using the Normal Distribution

Once a z-score is computed, the standard normal distribution (or Z-distribution) can be used to determine the probability of observing a sample mean as extreme as, or more extreme than, a specified value. This process allows for quantifying the likelihood of events under the assumption of normality, essential for inferential statistics.

Central Limit Theorem

The Central Limit Theorem states that the distribution of sample means will tend to be normal or approximately normal if the sample size is sufficiently large, regardless of the shape of the population distribution. This principle is crucial for making probability statements about the sample mean when using the normal distribution method for inference.

Sampling Distribution of the Mean

The sampling distribution of the mean describes how the means of random samples of a fixed size are distributed around the population mean. It forms the basis for understanding the variability of the sample mean and for performing hypothesis tests or confidence interval estimations regarding the population mean.

Transcript

00:01 According to this problem, the average movie ticket in the second quarter of 2015 was $8 .61.

00:12 So here we're talking about the population of movie tickets.

00:24 We also know from the given information that the population standard deviation is $1 .39.

00:39 And in this problem, we are going to do a random sample of 50 movie tickets.

00:45 Population we are drawing a sample and the sample size is going to be 50 tickets and the question is asking us what is the probability that the mean movie ticket of these 50 that we have selected is going to be exceeding which is the same thing is greater than eight dollars so because we are talking about a sample and the fact that we are dealing with a probability involving the average being greater than $8, we are applying the central limit theorem.

01:28 And with the central limit theorem, we know that the average of the means is going to be the same as the average of the population.

01:39 And in this case, it's $8 .61.

01:42 We also know the standard error of the mean, or the standard deviation of the means, is equal to the standard deviation of the population divided by the square root of n and in this case that would be a dollar 39 divided by the square root of 50 now we're going to have to draw a bell -shaped curve to represent the situation going on so our bell -shaped curve is going to have the average at the center and our average is eight dollars and sixty -one and we are trying to determine the probability that the average is greater than eight.

02:31 So $8 would be somewhere to the left of that, and we are talking going greater than.

02:38 So the next thing we would have to do is we would have to calculate the z score associated with $8...

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