117 7.9 Topic 9: Sampling Distributions 71. In a few sentences, describe what the sampling distribution of the mean represents in your own words. 12. Im 2012, MetricNet found that the average legth of a phone call to a help desk was about tem minutes. Call times were strongly right skewed with a standard deviation around 3.5 minutes. Suppose we take a random sample of 50 calls to the help desk and find the average () Cat duration for the 50 calls. We repeat this process many times. (a) What do you expect the mean of the 's to be? (b) What do you expect the standard deviation of the 's to be? (c) What do you expect the shape of the z's to be? What guarantees this fact? 73. The on-campus housing costs for a semester at a state university are Normally distributed with a mean of $1553 and a standard deviation of $329. (a) What is the distribution of the mean on-campus housing cost for a semester at this university for a random sample of 50 students? (b) What is the probability that a randomly selected on-campus student spends more than $1200 per semester? (c) What is the probability that a random sample of 50 students spends an average of less than $1500? (d) What is the probability that a random sample of 50 students spends an average of $1450 to $1600 per semester? 74. Sunshine City was designed to attract retired people, and its current population of 50,000 residents has a mean age of 60 years and a standard deviation of 16 years. The distribution of ages is skewed to the left. (a) What is the distribution of the sample mean age of residents for a random sample of 100 residents? (b) What is the probability that a random sample of 100 residents have a mean age between 50 and 70 years of age? (c) What is the probability that the mean age of a random sample of 100 residents is less than 55? 75. According to Quantcast analytics of CrossFit.com, 59% of CrossFitters have children. Suppose we take a random sample of 40 Crossfitters and find the sample proportion (p) that have children. We repeat this process many times. (a) What do you expect the mean of the p's to be? (b) What do you expect the standard deviation of the p's to be? (c) What do you expect the shape of the p's to be? 76. According to Quantcast analytics of CrossFit.com, 40% of CrossFitters have post-graduate de- grees. Suppose we take a random sample of 60 Crossfitters. (a) Describe the sampling distribution of the sample proportion of CrossFitters who have post- graduate degrees for samples of size 60.

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.