Rainfall Statistics from Cornell's Northeast Regional Climate Center indicate that Ithaca, New Y

Rainfall Statistics from Cornell's Northeast Regional...

Key Concepts

Normal Distribution

A type of continuous probability distribution for a real-valued random variable. It is symmetric about its mean and characterized by its bell-shaped curve. The distribution is defined by two parameters: the mean (which centers the distribution) and the standard deviation (which measures the spread of the data). This model is widely used in statistics due to its natural occurrence in many real-world phenomena.

Z-Score

A measure that indicates how many standard deviations an element is from the mean of a distribution. By converting observations to Z-scores, one can standardize different normal distributions and use standard normal tables to calculate probabilities and percentiles, facilitating comparisons across different datasets.

Percentiles and Tail Probabilities

Concepts related to the position of a value within a probability distribution. Percentiles indicate the value below which a given percentage of observations in a distribution falls. Tail probabilities refer to the probability that an observation falls in the extreme ends (either high or low) of the distribution, which is useful for identifying uncommon or rare events.

Sampling Distribution of the Sample Mean

The probability distribution of all possible sample means from a population. When samples are taken from a population (especially one that is normally distributed), the distribution of the sample mean is itself normally distributed, with a mean equal to the population mean and a standard deviation (standard error) equal to the population standard deviation divided by the square root of the sample size.

Central Limit Theorem

A fundamental theorem in statistics which states that, regardless of the original distribution of the data, the sampling distribution of the sample mean will tend to approximate a normal distribution as the sample size becomes larger. This theorem is central to making inferences about population parameters using sample statistics.

Standard Error

The standard deviation of the sampling distribution of a statistic, commonly the sample mean. It quantifies the dispersion of the sample mean around the population mean and is calculated as the population standard deviation divided by the square root of the sample size. A smaller standard error indicates a more precise estimate of the population mean.

Transcript

00:01 In problem 52 the amount of rain in ithaka follows a normal distribution with an average equals 35 .4 inches of each year and a standard deviation of 4 .2 inches for part a we want to calculate the percentage that ithaca get more than 40 inches of rain we want to calculate the percentage of years where ithaca gets more than 40 inches of rain.

00:33 First let's denote x as a random variable that represents the amount of rain each year in inches this means we need to calculate the proportion or the percentage for the random variable to be greater than 40 inches and this can be calculated using the z score or the standard normal probability the probability of z to be greater than the z score of 40 we can convert 40 to a z score by subtracting mu from it and divide by sigma there is mu and there is sigma then we need to calculate the probability for z to be greater than 1 .1 but we can calculate this probability directly we can calculate the probability for z to be smaller than a value because we have standard normal distribution tables that give us the value to the left of the z score if z equals 1 .1 here we can enter the tables and get the area to the left of the z score and of course this area represents the probability then we will write it as the total area under the curve which is 1 minus the probability for z to be smaller than 1 .1 now we can enter the table by the value of z equals 1 .1 to get the area to the left of 1 .1 .1 is given here and we take the first value because we want to get 1 .1010 the value is 0 .8 6433 then it equals 1 minus 4 .8 6433 it equals 0 .8 6433 it equals 0 .135 this is the percentage of year we can calculate it in percentage 13 .6 percent this is the percentage of years that has has rain more than 40 inches the amount of rain is greater than 40 inches for part b we want to calculate the amount of rain that is in the driest 20 percent of all years which means we have the probability for x to be smaller than a random value of x equals 0 .2 or 20%.

03:44 This is the value of the driest 20 % of all years because if we draw the standard the normal distribution at mu equals 35 .4, there is a value here that has a an area before it equals 0 .2 the area before it equals 0 .2 and we can get this value of x by equating this area to the standard area as done in part a we have a standard we have a standard normal distribution centered at z equal 0 and we have here a value of z that has an area before it equals 0 .2 can get x from z where x equals z multiplied by sigma plus mu and we can get z by entering the tables and search for the area equals 0 .2 and this area is to the left of the z score let's get the z score we search the area we search the tables for area equals 0 .2 there is an 0 .2 and this value is close to 0 .2 but this is more closer this is more closer than this value then we take z as minus 0 .8 and this column which is 0 1 2 3 4 then the value is minus 0 .84 which means x will be equal 0 .8 minus 0 .84 multiplied by sigma which is given as 4 .2 plus mu 35 .4 the answer equals 31 .872 inches this is the amount of rains that represent 20 % of years as the driest years for part c there is a university student in a faka for four years we represent by bar as the mean amount for those four years.

06:30 We want to describe that sampling distribution for y it follows a normal distribution as long as the population follows normal distribution and we know that x here follows a random distribution with mu equal 35 .4 and a variance of 4 .2 squared here y bar follows a normal distribution with the same mean as the population 35 .4 and a variance of the same variance divided by n or the same standard division divided by n all square square root of n which means we have the standard division 4 .2 of the population divided by square root of n square root of 4 all squared this means m for y bar equals 35 .4 and the standard deviation of y bar equals 2 .1 equals 2 .1 inches...

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