Question

1. Write a brief but complete discussion in which you cover the following topics: What is the mean  x and standard deviation  x of the x distribution based on a sample size n? Be sure to give appropriate formulas in your discussion. How do you find a standard z score corresponding to x ? State the central limit theorem and the general conditions under which it can be used. Illustrate your discussion using examples from everyday life.

i. According to the examination board, 48% of all the voters in the district support the referendum. Suppose that a principal is interested in the proportion of voters who support the referendum in a group of 38 caretakers. Approximate p ˆ by a normal Distribution

Answer 1

ANSWER::

The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30.

Conditions for CLT :-
Randomization Condition: The data must be sampled randomly.
10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population.
Independence: The sample values must be independent of each other. This means that the occurrence of one event has no influence on the next event.

Example can be like:-
A population of 29 year-old males has a mean salary of $29,321 with a standard deviation of $2,120. If a sample of 100 men is taken, what is the probability their mean salaries will be less than $29,000?
Step 1: Insert the values into the z-formula:
= (29,000 – 29,321) / (2,120/√100) = -321/212 = -1.51.
Step 2: Look up the z-score in the left-hand z-table (or use technology). -1.51 has an area of 93.45%.
However, this is not the answer, as the question is asking for LESS THAN, and 93.45% is the area “greater than” so you need to subtract from 100%.
100% – 93.45% = 6.55% or about 0.07.

i.

If repeated random samples of a given size n are taken from a population of values for a quantitative variable, where the population mean is μ (mu) and the population standard deviation is σ (sigma) then the mean of all sample means (x-bars) is the population mean μ (mu).

Since the square root of sample size n appears in the denominator, the standard deviation does decrease as the sample size increases.

Mean = p = 0.48

Sample size = 38

The standard deviation = 0.48*0.52/38 = 0.08

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