7.27 Suppose that we observe a random variable having the binomial distribution. Let X be the number of successes in n trials.

a) Show that X n is an unbiased estimate of the binomial parameter p.

(b) Show that X + 1 n + 2 is not an unbiased estimate of the binomial parameter p.

7.28 The statistical program MINITAB will calculate the small sample confidence interval for μ. With the nanopillar height data in C1, Dialog box: Stat > Basic Statistics > 1-Sample t. Click on box and type C1. Choose Options. Type 0.95 in Confidence level and choose not equal. Click OK. Click OK produces the output N Mean StDev SE Mean 95% CI C1 50 305.580 36.971 5.229 (295.073, 316.087) The R command t.test(x,conf.level=.95) produces similar results when the data are in x.

(a) Obtain a 90% confidence interval for μ.

(b) Obtain a 95% confidence interval for μ with the aluminum alloy data on page 29. Alternatively, you can use the MINITAB commands Stat > Basic statistics > Graphical summary to produce the more complete output

Summary for height (nm) Anderson-Darling Normality Test A-Squared P-Value 0.38 0.398 Mean 305.58 StDev 36.97 Variance 1366.86 Skewness 0.260823 Kurtosis 0.202664 240 280 320 360 400 N 50 Minimum 221.00 1st Quartile 27750 Median 304.50 3rd Quartile 330.75 Maximum 391.00 95% Confidence Intervals Mean Median - 290 295 300 305 310 315 95% Confidence Interval for Mean 316.09 95% Confidence Interval for Median 311.33 295.07 292.00 30.88 95% Confidence Interval for StDev 46.07