Question

2) Decide whether the following statements are true or false. Explain your reasoning.

a) For a given standard error, lower confidence levels produce wider confidence intervals.

b) If you increase sample size, the width of confidence intervals will increase.

c) The statement, "the 95% confidence interval for the population mean is (350, 400)", is equivalent to the statement, "there is a 95% probability that the population mean is between 350 and 400".

d) To reduce the width of a confidence interval by a factor of two (i.e., in half), you have to quadruple the sample size.

e) Assuming the central limit theorem applies, confidence intervals are always valid.

f) The statement, "the 95% confidence interval for the population mean is (350, 400)" means that 95% of the population values are between 350 and 400.

g) If you take large random samples over and over again from the same population, and make 95% confidence intervals for the population average, about 95% of the intervals should contain the population average.

h) If you take large random samples over and over again from the same population, and make 95% confidence intervals for the population average, about 95% of the intervals should contain the sample average.

i)   It is necessary that the distribution of the variable of interest follows a normal curve.

j) A 95% confidence interval obtained from a random sample of 1000 people has a better chance of containing the population percentage than a 95% confidence interval obtained from a random sample of 500 people.

k) If you make go through life making 99% confidence intervals for all sorts of population means, about 1% of the time the intervals won't cover their respective population means.

Answer 1

Answer:

a) For a given standard error, lower confidence levels produce wider confidence intervals. FALSE

( Given that the standard error is fixed and decreasing the confidence level will decrease the width of the interval)

b) If you increase sample size, the width of confidence intervals will increase.- FALSE

( As the sample size increases , margin of error decreases which in turn decreases the width of the interval)

c) The statement, "the 95% confidence interval for the population mean is (350, 400)", is equivalent to the statement, "there is a 95% probability that the population mean is between 350 and 400"-FALSE

( As this statement means that we are 95% confident that the population mean lies between 350 and 400)

d) To reduce the width of a confidence interval by a factor of two (i.e., in half), you have to quadruple the sample size-  TRUE

( To reduce the width of a confidence interval by a factor of two we need to make the sample size from n to 4n as

is the factor that is mutiplied in the formula and to reduce the interval by a factor of 2 we need to make this )

e)False

Here 'valid' means that the confidence interval procedure has a 95% or 99%(what ever we assume) chance of producing an interval that contains the population parameter.

The central limit theorem is needed for confidence intervals to be valid.

f)False

Confidence interval does not provide a range for 95% of data values from the population. It refers to the percentage or probability that a population parameter will fall between 350 and 400 values for a certain proportion of time.

g)True

h)False.

The confidence interval is a range for the population average but not for the sample average. Moreover, every confidence interval has its corresponding sample average.

i.e, CI=sample average Standard deviation/Sample Size.

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