Suppose an approximate N(μ,σ) random variable describes the distribution of bank account balances in USD across...

Suppose an approximate N(μ,σ) random variable describes the distribution of bank account balances in USD across a population. Let this variable be X∼N(μ,σ). Suppose we knowσ= 1304 and we wish to estimateμ, the true mean bank account balance in USD across the population. We take a simple random sample of size 225 from the population and calculate the sample mean to be ̄x= 3178.a) Find a 98% confidence interval for μ

a.) Find a 98% confidence interval for μ.

b.) Suppose 100 other researchers have also produced similar confidence intervals (all oft hem use the same sample size of 225 but their samples are all different and independent of each other). How many of these confidence intervals would we expect to not contain the true population mean μ; i.e., how many of these confidence intervals are expected to be “bad” confidence intervals. Justify your answer.

c.) Suppose you wish to run a 2-sided hypothesis test at theα= 0.02 significance level with null hypothesis: H0:μ = 3300 using the same data you used in your confidence interval. Based on your confidence interval result, would you reject or not reject the null hypothesis? Do NOT run the hypothesis test itself, state your conclusion based on the confidence interval and briefly summarize why the test and confidence interval are related.

Solutions

Expert Solution

population std dev ,    σ =    1304.0000
Sample Size ,   n =    225
Sample Mean,    x̅ =   3178.0000
Level of Significance ,    α =    0.02
'   '   '
z value=   z α/2=   2.3263   [Excel formula =NORMSINV(α/2) ]

Standard Error , SE = σ/√n =   1304.0000   / √   225   =   86.933333
margin of error, E=Z*SE =   2.3263   *   86.93333   =   202.237175

confidence interval is
Interval Lower Limit = x̅ - E =    3178.00   -   202.237175   =   2975.76282
Interval Upper Limit = x̅ + E =    3178.00   -   202.237175   =   3380.23718
98%   confidence interval is (   2975.76   < µ <   3380.24   )

...

Ho : µ = 3300
Ha : µ ╪ 3300

98% confidence interval is ( 2975.76 < µ < 3380.24 )

3300 lie with in the CI , so null hypothesis will not be rejected.

...................

Please revert back in case of any doubt.

Please upvote. Thanks in advance.

orchestra answered 1 year ago

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