Cloudiness in Breslau. In the paper “Cloudiness: Note on a Novel Case of Frequency” (Proceedings of the Royal Society of London, Vol. 62), K. Pearson examined data on daily degree of cloudiness, on a scale of 0 to 10, at Breslau (Wroclaw), Poland, during the decade 1876–1885. A frequency distribution of the data is presented in the following table. From the table, we find that the mean degree of cloudiness is 6.83 with a standard deviation of 4.28. Degree Frequency Degree Frequency 0 751 6 21 1 179 7 71 2 107 8 194 3 69 9 117 4 46 10 2089 5 9 a. Consider simple random samples of 100 days during the decade in question. Approximately what percentage of such samples have a mean degree of cloudiness exceeding 7.5? b. Would it be reasonable to use a normal distribution to obtain the percentage required in part (a) for samples of size 5? Explain your answer.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
Cloudiness in Breslau. In the paper “Cloudiness: Note on a Novel Case of Frequency” (Proceedings of the Royal Society of London, Vol. 62), K. Pearson examined data on daily degree of cloudiness, on a scale of 0 to 10, at Breslau (Wroclaw), Poland, during the decade 1876–1885. A frequency distribution of the data is presented in the following table. From the table, we find that the
| Degree | Frequency | Degree | Frequency |
| 0 | 751 | 6 | 21 |
| 1 | 179 | 7 | 71 |
| 2 | 107 | 8 | 194 |
| 3 | 69 | 9 | 117 |
| 4 | 46 | 10 | 2089 |
| 5 | 9 |
a. Consider simple random samples of 100 days during the decade in question. Approximately what percentage of such samples have a mean degree of cloudiness exceeding 7.5?
b. Would it be reasonable to use a
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