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Does asymptotic mean that the normal curve gets closer and closer to the X-axis but never actually touches it? Yes, asymptotic means that the curve of a line will approach 0 (the x-axis), but it will not touch 0 and instead will extend to infinity. In this class, this applies to the normal continuous distribution and is one of the 4 key characteristics of a normal continuous distribution that our text book discusses. This means that the curve of the line will extend infinitely in both the negative and positive direction in exact mirror image patterns on either side of the mean.

For a normal probability distribution, is about 95 percent of the area under normal curve within plus and minus two standard deviations of the mean and practically all (99. 73 percent) of the area under the normal curve is within three standard deviations of the mean? Yes. According to the Empirical Rule: -68% of the area under the curve is within +/- 1 standard deviation of the mean -95% of the area under the curve is within +/- 2 standard deviations of the mean -Virtually all, 99.

% of the area under the curve is within +/- 3 standard deviations of the mean Is a z-score the distance between a selected value (X) and the population mean (u) divided by the population standard deviation(s)? Yes. We use z-scores to change normal probability distributions into standard normal probability distributions, which are unique because they have a mean of 0 and standard deviation of 1. To convert to a standard normal probability distribution we must find the z-scores for each observation.

These are found by subtracting the mean value from the selected value and dividing by the standard deviation. The Normal Probability Distribution Find an example of application of probability theory in your workplace or business. Show that the reasons that your workplace uses probability analysis, such as probability of risk calculations or percent defects or percent for pass or fail of a product, etc. In my company, I do groundwater sampling for remediation projects. When we are finished, we send our samples to a laboratory via FedEx or UPS.

The laboratory reports that approximately 2 bottles are broken in every cooler shipped, regardless of how well they are packed. To perform sample analysis, the laboratory needs 1-500 ml bottle of groundwater, and 1-50ml vial of water to perform all of the tests for each well. When we take samples we collect 3-500ml bottles and 3-50 ml vials of groundwater per well because we know that on average two bottles will break per shipment. The bottles that break could be from 2 different wells, or 2 different sized bottles, or they could be two identical sized bottles from the same well.

By collecting extra samples, we ensure that we are sending the lab enough samples to accurately perform analysis, and we are ensuring that we don’t have to go back into the field and spend thousands of extra dollars to re-collect samples. What are some of characteristics of a Normal Probability Distribution? According to our text (pg 223), all normal probability distributions have these characteristics: 1. The are bell-shaped and the mean, median, and mode are equal and located in the centre of the distribution. 2. The total area under the curve = 1. 00 with ? f this located to the right of the peak(mean) and ? located to the left of the peak (mean). 3. The distribution curve is symmetrical around the peak (mean) and therefore there are two identical halves of the curve, centred around the mean. 4. The curve approaches the x-axis, but never actually touches it. (i. e. , it is asymptotic) 5. The location is determined by the mean and the dispersion is determined by the standard deviation. Non-stop Airlines determined that the mean number of passengers per flight is 152 with a standard deviation of ten passengers.

Practically do all flights have between 142 and 162 passengers? According to the Empirical rule, 142 -162 passengers would fall within 1 standard deviation of the mean (i. e. , 68% of the area under the curve) If we wanted to know how many passengers were on practically/virtually all flights, we would have to apply the Empirical Rule for 3 standard deviations from the mean. This would account for 99. 7% of the area under the curve. According to this theory, virtually all flights would have between 122 – 182 passengers.

Is the total area within any continuous probability distribution equal to 1. 00? Yes. If we are a talking about uniform probability distributions (rectangles), the area must equal 1. We can find this using Area = basexheight or (b-a/1) x (1/b-a). Using this equation, both fractions will ‘cancel out’ to give you a value of 1. 00. If we are talking about normal probability distributions, they are bell-shaped with a single peak at the distribution centre and therefore, they are symmetrical about the mean.

This means that the two halves of the curve are identical and they both have values of 0. 5 (0. 5 to the left of the mean and 0. 5 to the right of the mean). Is the uniform probability distribution’s standard deviation proportional to the distribution’s range? Yes. The equation for standard deviation for a uniform probability distribution is = SQRT [ (b-a)^2/12]. A range is the difference between the max and min values for a distribution (b-a).

Therefore, the range of the distribution directly impacts the standard deviation as it is a part of the equation. The larger the range, the larger the standard deviation of a uniform distribution and the smaller the range, the smaller the standard deviation of a uniform distribution. About what percent of the area under the normal curve is within one standard deviation of the mean? According to the Empirical Rule, approximately 68% of the area under the curve, for a normal distribution, is within +/- one standard deviation of the mean. (u +/- 1sd)

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