samples of size n=900 are randomly selected from the population of numbers (0 through 9) produced by a random-number generator, and the proportion of numbers greater than 5 is found for each sample. what is the shape distribution of the sample proportions
It’s normal approximately
The distribution of sample proportions of a binomial distribution follows an approximately normal distribution under certain conditions.
In this case, we have a population of numbers produced by a random-number generator, where each number has a probability of success (greater than 5) of 40% and a probability of failure (less than or equal to 5) of 60%. This satisfies the conditions for the Central Limit Theorem.
According to the Central Limit Theorem, if the sample size is large enough (in this case n=900), the distribution of sample proportions will be approximately normal. The mean of the sample proportions will be equal to the population proportion (40% in this case), and the standard deviation of the sample proportions will be given by the formula:
Standard deviation = √[(p * q) / n]
where p is the proportion of success (40% = 0.4), q is the proportion of failure (60% = 0.6), and n is the sample size (900 in this case).
Therefore, the shape of the distribution of sample proportions in this scenario is approximately normal.
To determine the shape distribution of the sample proportions, you need to consider the properties of the sampling distribution.
The sampling distribution of sample proportions is well-approximated by a normal distribution when certain conditions are met. One of these conditions is that the sample size must be large enough.
In this case, the sample size is n = 900, which is considered large enough for the sampling distribution to be approximately normal. This is due to the Central Limit Theorem, which states that as the sample size increases, the sampling distribution tends to follow a normal distribution regardless of the shape of the population.
Therefore, the shape distribution of the sample proportions, in this case, can be approximated by a normal distribution.