A) Choose one of the following distributions (positively skewed normal distribution, negatively skewed normal distribution or bimodal normal distribution) and generate a sample of 20 numbers for it. The sample you generate should have a mean of 8 and a sample standard deviation of 4.
To generate a sample of 20 numbers for a specified distribution, we need to understand each of the distributions mentioned: positively skewed normal distribution, negatively skewed normal distribution, and bimodal normal distribution.
1) Positively skewed normal distribution:
In a positively skewed normal distribution, the data is concentrated towards the lower values but has a long tail on the right side. To generate a sample with these characteristics, we can make use of the inverse transform method:
a) Generate a set of 20 random numbers from a standard normal distribution. We can use any programming language or statistical software to do this. Let's denote this set as Z.
b) Apply the inverse cumulative distribution function (CDF) of a right-skewed distribution such as the chi-square distribution or the log-normal distribution on each element of Z. This will transform the values to follow a positively skewed distribution.
c) Adjust the transformed values to have the desired mean and sample standard deviation. This can be done by multiplying each transformed value by the desired standard deviation and adding the desired mean.
2) Negatively skewed normal distribution:
In a negatively skewed normal distribution, the data is concentrated towards the higher values but has a long tail on the left side. We can use a similar approach as for the positively skewed distribution:
a) Generate a set of 20 random numbers from a standard normal distribution, denoted as Z.
b) Apply the inverse cumulative distribution function (CDF) of a left-skewed distribution, such as a negative chi-square distribution or a reciprocal transformation to each element of Z, transforming the values to follow a negatively skewed distribution.
c) Adjust the transformed values to have the desired mean and sample standard deviation by multiplying each transformed value by the desired standard deviation and adding the desired mean.
3) Bimodal normal distribution:
In a bimodal normal distribution, the data has two distinct peaks. To generate a sample with these characteristics:
a) Generate two sets of 10 random numbers from independent standard normal distributions, denoted as Z1 and Z2.
b) Adjust each set of random numbers to have a desired mean and standard deviation. You can multiply each set by the desired standard deviation and add the desired mean.
c) Combine the two sets of adjusted random numbers to form a sample of 20 numbers. This will create a bimodal distribution.
Keep in mind that the exact formulas and procedures may vary depending on the programming language or statistical software that you are using.