Question

A new radar device is being considered for a certain missile defense system. The system is checked by experimenting with aircraft in which a kill or a no kill is simulated. Two research group A and B participated the simulation, and it was found that 63 kills occur out of 100 trials by group A, and 59 kills occur out of 125 trials by group B. Is there a significant difference between the proportions of group A and B's simulation? Use P-value with significance level of 0.05 to draw a conclusion.

Answer 1

To determine if there is a significant difference between the proportions of group A and B's simulations, we can perform a hypothesis test using the z-score. We calculate the proportions for each group, then the standard error and z-score. Finally, we compare the p-value to the significance level of 0.05 to conclude.

Step-by-step explanation:

To determine if there is a significant difference between the proportions of group A and B's simulations, we can perform a hypothesis test. The null hypothesis (H0) states that there is no significant difference between the proportions, while the alternative hypothesis (H1) states that there is a significant difference. The test statistic for this problem is the z-score.

Step 1: Calculate the proportions for each group. For group A, the estimated proportion is 63/100 = 0.63. For group B, the estimated proportion is 59/125 = 0.472.

Step 2: Calculate the standard error using the formula: sqrt((p₁(1-p₁)/n₁) + (p₂(1-p₂)/n₂)), where p₁ and p₂ are the proportions of each group, and n₁ and n₂ are the sample sizes. Step 3: Calculate the z-score using the formula: (p1 - p₂) / SE, where SE is the standard error.

Step 4: Find the p-value associated with the z-score. This can be done using a standard normal distribution table or a statistical software.

Step 5: Compare the p-value to the significance level of 0.05. If the p-value is less than 0.05, we reject the null hypothesis. If the p-value is greater than or equal to 0.05, we fail to reject the null hypothesis.

Answer 2

There is significant difference between the proportions of group A and group B.

Step-by-step explanation:

In this question we need to perform an hypothesis test on the difference of proportions (Δπ) and determine if there is any significant difference between the two proportions.

The null and alternative hypothesis are:

`H_0: \Delta \pi=0\\\\H_1: \Delta \pi\\eq0`

The proportion of group A is
`p_1=63/100=0.630`.

The proportion of group B is
`p_2=59/125=0.472`.

The weighted average of p (to calculate s) is:

`p=(n_1*p_1+n_2*p_2)/(n_1+n_2)=(100*0.630+125*0.472)/(100+125)=(63+59)/(100+125)=(122)/(225)=0.542`

The estimated standard deviation is:

`s=\sqrt{(p(1-p))/(n_1)+(p(1-p))/(n_2) } =\sqrt{(0.542(1-0.542))/(100)+(0.542(1-0.542))/(125)} =\sqrt{(0.248)/(100) +(0.248)/(125) } \\\\s=√(0.00447) =0.067`

Then we can calculate z as

`z=(p_1-p_2)/(s)=(0.630-0.472)/(0.067)=(0.158)/(0.067)=2.36`

The P-value for z=2.36 is 0.009, which is less that the significance level of 0.05. The effect is significant and the null hypothesis is rejected.

There is significant difference between the proportions of group A and group B.