Violence and stress. Interpersonal violence (e.g., rape)...

Violence and stress. Interpersonal violence (e.g., rape) generally leads to psychological stress for the victim. Clinical Psychology Review (Vol. 15, 1995) reported on the results of all recently published studies of the relationship between interpersonal violence and psychological stress. The distribution of the time elapsed between the violent incident and the initial sign of stress has a mean of 5.1 years and a standard deviation of 6.1 years. Consider a random sample of $n=150$ victims of interpersonal violence. Let $\bar{x}$ represent the mean time elapsed between the violent act and the first sign of stress for the sampled victims. a. Give the mean and standard deviation of the sampling distribution of $\bar{x}$. b. Will the sampling distribution of $\bar{x}$ be approximately normal? Explain. c. Find $P(\bar{x}>5.5)$. d. Find $P(4<\bar{x}<5)$.

Violence and stress. Interpersonal violence (e.g., rape) generally leads to psychological stress for the victim. Clinical Psychology Review (Vol. 15, 1995) reported on the results of all recently published studies of the relationship between interpersonal violence and psychological stress. The distribution of the time elapsed between the violent incident and the initial sign of stress has a mean of 5.1 years and a standard deviation of 6.1 years. Consider a random sample of $n=150$ victims of interpersonal violence. Let $\bar{x}$ represent the mean time elapsed between the violent act and the first sign of stress for the sampled victims.
a. Give the mean and standard deviation of the sampling distribution of $\bar{x}$.
b. Will the sampling distribution of $\bar{x}$ be approximately normal? Explain.
c. Find $P(\bar{x}>5.5)$.
d. Find $P(4<\bar{x}<5)$.

Key Concepts

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, most commonly the sample mean. It is calculated as the population standard deviation divided by the square root of the sample size, and it quantifies the variability of the sample mean from one sample to another.

Standardization (Z-score Calculation)

Standardization involves converting a sample observation or a sample mean into a Z-score by subtracting the mean and dividing by the standard error. This process allows one to use the standard normal distribution to determine probabilities and critical values related to the sampling distribution.

Sampling Distribution

The sampling distribution is the probability distribution of a statistic, such as the sample mean, that is obtained by repeatedly sampling from a population. It reflects how the sample statistic varies from sample to sample and is a foundational concept for making inferences about a population from a sample.

Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the shape of the original population distribution. This theorem justifies using normal probability calculations even when the population distribution is not normal, provided the sample size is sufficiently large.

Normal Probability Calculations

Normal probability calculations involve determining the probability of a statistic falling within a specific range using the properties of the normal distribution. This method relies on the computation of Z-scores and subsequent lookup or calculation of probabilities, which is essential for hypothesis testing and confidence interval estimation.

Transcript

00:01 So we have the mean time of the anxiety being 5 .1 years after the event and with a standard deviation of 6 .1.

00:13 I got to double check and make sure that that's not 6 .7.

00:15 Yes, 6 .1 years.

00:18 And we're taking a sample of 150 women.

00:22 And we want to know for the sampling distribution what will be the mean of a sampling distribution, which that would be 5 .1 years.

00:29 And the standard deviation of that would end up being the 6 .1 divided by the square root of 150.

00:38 And 6 .1 divided by square root 150 comes out to be almost 0 .45, so 0 .498 or approximately 0 .5 years, half a year.

00:50 And in part b, they ask us, will the sampling distribution be approximately normal and why? and the answer is yes, it will be approximately normal.

01:01 And why? because the central limit theorem tells us that if we take random samples of large populations and large is kind of deemed, if it's usually greater than or equal to 30, we're good to go, that the sampling distribution of x bars will tend to follow very close to a normal distribution.

01:20 So yes, it is going to be approximately normal.

01:25 Then in part c, we want to find what's the likelihood that one of those means of 150 is higher than 5 .5 years.

01:34 And so let's find that, convert it to a z.

01:37 And again, here's our little picture, which i always like to draw a little picture.

01:43 I'm doing the little picture in my mind if i don't draw it out on paper.

01:47 And this standard deviation is about 0 .5.

01:49 So this is going to end up being 5 .6.

01:52 This is going to be 6 .1.

01:53 Back here will be 4 .6 and so we know that the 5 .5 is going to be a little less than one standard deviation...