Final answer:
The normal distribution discussed has a mean, mode, and median all at the same value. When the standard deviation increases, the curve widens. Understanding z-scores, sampling distributions, and hypothesis testing are crucial when working with normally distributed data.
Step-by-step explanation:
Understanding Normal Distributions
The topics being discussed are related to the properties of a normal distribution in statistics, a fundamental concept in probability theory and statistics. A normal distribution is a bell-shaped curve that is symmetrical about the mean (μ). For a normal distribution:
When considering the mean of 22 and a standard deviation of 3 mentioned in the student's question, both the mean and mode are indeed 22. If the standard deviation is increased to 4, as per the student's hypothesis, the curve will become wider, representing increased variability in the data values around the mean.
In a classroom setting, when plotting the heights of male and female students separately on a histogram and then drawing a smooth curve, if the sample size were large enough, we would expect the curves to approximate a normal distribution. The mean of the distribution would be indicated on the x-axis, and calculating the probability of heights above or below a certain value would involve shading the area under the curve beyond that value.
Regarding hypothesis testing, when we have a population mean, a sample mean, a sample standard deviation, and if the underlying population is assumed to be normal, one typically uses a t-distribution for hypothesis testing when the sample size is small. However, if the sample size is large (generally n > 30), the sampling distribution of the sample mean can be approximated by a normal distribution due to the Central Limit Theorem.
Finally, the mean of a sampling distribution of the sample mean would be the population mean, and its standard deviation would be the population standard deviation divided by the square root of the sample size. When referring to z-scores in the context of a standard normal distribution, we indicate a mean of zero and a standard deviation of one.