053. Using the Sample Distribution

Samples have a very direct and consequential impact on decisions that are made. An extremely common and quite useful application of a sampling distribution is to determine the probability that a sample mean will fall within a given range. Given that the sampling distribution will be normally distributed because (1) the sample is taken from a normal population or (2) N ≥ 30 and the Central Limit Theorem ensures normality in the sampling process.

There is a Conversion formula for determining the probability of selecting one observation that would fall within a given range

,

Where X is a single observation of interest; σ is the population standard deviation.

However, many business decisions depend on an entire sample – not just one observation. In this case, the conversion formula must be altered to account for the mean of several observations, . Therefore, when sampling is done, the conversion formula becomes

, (5.8)

Where is the mean of several observations; is the standard error of the sampling distribution.

Z – Formula has the Purpose: It is used to convert all normal distributions of to a standard form.

Example 5.2. The Telcom recorded telephone messages for its customers. These messages averaged 150 seconds with a standard deviation of 15 seconds. Telcom wished to determine the probability that the mean duration of a sample of N = 35 phone calls is between 150 and 155 seconds, that is P(150 < < 155).

Solution. Since N > 1 and a sample was taken, Formula (5.8) must be used. Then

or an area of 0.4756.

Thus, P(150 < < 155) = P(0 < Z< 1.97) = 0.4756. See it in Figure 5.3. so the probability that a sample of 35 calls will have duration within the range of 150 and 155 seconds is 47.56 percent. Such a quite big percentage is because the sampling distribution is less dispersed than the original population, i. e. the dispersion of original population is bigger than dispersion of the sampling distribution ( > ).

Figure 5.3 – The Mean of a Sample Observations

By being able to predict the likelihood that a certain statistic will fall within a given range, decision making becomes more precise and scientific.

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