Question

1 a) Approximately 95% of all values of a normally distributed populations lie within how many standard deviations of the population mean

b) A random sample of ten items is taken from a population. The items have the following values:  

9 32 37 32 21 24 10 15 21 30. Find the point estimate for the confidence interval

c) State what effect on the margin of error for a confidence interval each of the following will have.

i) Increase sample size while keeping the same confidence level

ii) Increase the confidence level while keeping the same sample size

iii) Decrease confidence level and increase sample size.

d) If a sample mean is 52.8 and the confidence interval has a margin of error of 3.4. State the confidence interval

e) If you were working with a confidence level of 95% and you took 100 different samples to develop one hundred different confidence intervals. Approximately How many of those confidence intervals should contain the true population mean

f)Two events that cannot happen at the same time are referred to as?

Answer 1

1) Approximately 95% of all values of a normally distributed populations lie within Two standard deviations of the population mean.

( according to the empirical rule)

2) Not Possible to calculate.

( As we know that confidence interval is an interval and point estimate is a single point . So we cannot get a point estimate of an interval )

3) Note - Margin of error = $z \frac{\sigma }{\sqrt{n}}$

i) DECREASE THE MARGIN OF ERROR

( as we can see from the formula that the sample size n is in denominator, That means increasing n will decrease the margin of error )

ii) INCREASE THE MARGIN OF ERROR

( as we know that increasing the confidence level, the interval become more wider, so that means that margin of error increases)

iii) DECREASE THE MARGIN OF ERROR

( as, increase in the sample size will decrease the margin of error , Plus the decrease in the confidence level will also decrease  the margin of error )

d) sample mean , $\bar{x}$ = 52.8

Margin of error, E = 3.4

Confidence interval = $\bar{x}\pm E$

= ( $52.8\pm3.4$ )

= ( 49.4, 56.2)

e) Approximately 95 of these confidence interval should contain the true population mean

f) Two events that cannot happen at the same time are referred as Mutually excusive events