Use http://www.lock5stat.com/StatKey/ Restaurant tips dataset include tips left

Question

Use http://www.lock5stat.com/StatKey/

Restaurant tips dataset include tips left by a random sample 157 customers in a New York bistro. Sample mean for this data is 3.849 and the sample standard deviation is 2.421.

(This dataset is listed under the bootstrap confidence intervals for mean, median, st.dev in StatKey)

a.    If we are interested in estimating the true mean amount of tips leave by customers in this restaurant, describe how you would use the original sample to get a one bootstrap sample. What statistic is recorded from the sample?

b.    What shape and center do we expect for the bootstrap distribution?

c.    Find 95% confidence interval for true mean using:

                                    i.    One sample t-interval

                                   ii.    Bootstrap z-interval (use 3000 bootstrap samples)

                                  iii.    Bootstrap percentile interval (use 3000 bootstrap samples)

d.    Find a 90% confidence for true standard deviation of the tips.

Explanation / Answer

a) For bootstrapping we will use simple random sampling on iterative basis. We can keep drawing samples with a sample size and noting their mean and std deviation.

b) This bootstrap sampling will lead us to follow notion of Central Limit Theorem (CLT). The moment we draw the distribution of samples' mean, we would expect a Normal distribution with centre as a grand mean of population and std deviation. It will symmetry about its mean.

c) we compute standard error SE=

std dev/sqrt(n) = 0.193

i) CI for mean using t-dist= mean+-t(.975)*SE = [3.467, 4.231]

Excel formula: =3.849 +- T.INV.2T(0.05,156)*.19

ii) CI for mean using t-dist= mean+-z(.975)*SE = [3.47, 4.227]

Excel formula: =3.849 +- NORM.S.INV(0.975)*.19

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