Question

=(54.3 ± 2.756(125)( 48.010, 60.590) (6. An Asian roulette wheel has 38 numbers; 18 red, 18 black, and 2 green. perfectly balanced roulette wheel, red numbers should turn up 18 times in 38 spins of the wheel. To test the integrity of the wheel, Luxor Casino records the resultss of 3,801 plays. Of the 3,801 plays, a red number appeared 1,888 times. Luxor Casino suspects With a that the roulette wheel favors red numbers more often. i. Formulate the null and alternative hypotheses. Ho : p38 0.474 = 0.474 vs. H: p> ii. Compute the test-statistic and p-value. test= 0.497-0.474 2.840. Then V(0.474 0.526)/3801 p-value P(Z> ztest) 0.0023 iii. What is your decision and what do you conclude? (Use a 0.10) Since 0.0023 < 0.10 then we decide to reject Ho. We conclude, at a there is enough evidence to reject the claim that the proportion of red numbers observed when playing the Asian roulette wheel is Furthermore, that the proportion of red numbers observed when playing the Asian roulette wheel is larger than stated. 10% level of significance, 18 38 conclude we iv. Construct a 90% confidence interval for the true proportion of red numbers that should appear. (0.484, 0.510)

Answer 1

6.

i.

H0: p $\leq$ 0.474

H1: P > 0.474 (one tailed - right tailed test)

ii.

Sample proportion, $\hat{p}$ =1888/3801 =0.497

Hypothesised proportion, p =0.474

Standard Error, SE =
Ур(1 - р) /п V
=
V0.474(1 0.474)/3801
=0.0081

Test statistic, Z =($\hat{p}$ - p)/SE =(0.497 - 0.474)/0.0081 =2.84

p-value for Z =2.84 for one-tailed test is: p-value =0.0023

iii.

Decision:

Since p-value of 0.0023 < $\alpha$ of 0.10, we reject the null hypothesis(H0) at 10% significance level in favor of the alternative hypothesis(H1).

Conclusion:

There is a sufficient statistical evidence to claim that the proportion of red numbers is greater than 0.474 (i.e., >18/38).

iv.

At 90% confidence level, for a right-tailed test, the critical value of Z is: Z_crit =1.28

Margin of Error, MoE =Z_crit*SE =1.28*0.0081 =0.0104

One sided 90% confidence interval for the true proportion (p) of red numbers is:

Lower limit for p = -$\propto$

Upper limit for p =$\hat{p}$ + MoE =0.497 + 0.0104 =0.507

Thus, one sided 90% confidence interval for p =(-$\propto$​​​​​​, 0.507)