Answer:
- The required percentage to 4 d.p. = 59.5126%
- The required percentage in decimal point to 4 d.p. = 0.5951
Step-by-step explanation:
Proportion that said they have enough money to live comfortably now and expected to do so in the future = 55% = 0.55
Sample size = n = 200
The Central Limit Theorem ensures that we can say that:
1) The sample proportion is equal to the population proportion.
p = μₓ = μ = 0.55
2) Standard Deviation of the sampling distribution = σₓ = √[p(1-p)/n] = √(0.55×0.45/200) = 0.0352
3) We can say that the sample distribution approximates a normal distribution especially when
np > 5 and nq > 5 (which is true for this sample size)
The probability is 90% that less than what sample percentage will say they expect to live comfortably
To find this sample percentage, let that that sample percentage be x' and its z-score be z'
z' = (x' - μ)/σₓ
But we are told that
P(z < z') = 90% = 0.90
Using the normal distribution table
z' = 1.282
1.282 = (x' - 0.55)/0.0352
x' = 0.55 + (0.0352×1.282) = 0.5951264
Hence, the required sample percentage = 59.5126% to 4 d.p.
Hope this Helps!!!
