A new radar device is being considered for a certain missile...

Name: Problem 57, Chapter 10: Probability & statistics for engineers & scientists
Uploaded: 2021-10-29T05:20:17-08:00
Duration: 5 min 21 s
Channel: Ajiboye Tunde
Description: 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. If, in 300 trials, 250 kills occur, accept or reject, at the 0.04 level of significance, the claim that the probability of a kill with the new system does not exceed the 0.8 probability of the existing device.

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. If, in 300 trials, 250 kills occur, accept or reject, at the 0.04 level of significance, the claim that the probability of a kill with the new system does not exceed the 0.8 probability of the existing device.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical process used to evaluate a claim about a population parameter by comparing sample data against a null hypothesis. This method helps to decide whether any observed differences are attributable to chance or if there is enough evidence to support a specific alternative claim.

Null and Alternative Hypothesis

The null hypothesis represents a baseline or status quo claim about the population parameter, while the alternative hypothesis is what one seeks to support with evidence from the sample. In proportion testing, the null hypothesis might state that the probability of success is equal to or exceeds a given benchmark, and the alternative hypothesis would suggest that the probability is different or does not exceed that benchmark.

Significance Level

The significance level, denoted by alpha, is the threshold probability for making a Type I error—rejecting the null hypothesis when it is true. It defines the degree of risk the tester is willing to take, and comparisons of the p-value to this level determine the outcome of the hypothesis test.

Test Statistic

A test statistic is a standardized value calculated from sample data during a hypothesis test. In tests of proportions, this statistic often follows a normal distribution under the null hypothesis, and it quantifies the distance between the observed sample proportion and the null hypothesis proportion relative to the estimated standard error.

p-value

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed under the assumption that the null hypothesis is true. It provides a measure of the evidence against the null hypothesis; a p-value lower than the significance level indicates that the observed data are unlikely under the null hypothesis, leading to its rejection.

Transcript

00:01 So a missile is being tested and then we are asked to test the claim that the probability of killing with the new system does not exceed 0 .8 probability of the existing device.

00:12 And then 300 trials were made and 250 cues occurred.

00:19 Here is the worksheet.

00:20 We have our population proportion to be 0 .8.

00:23 We have our sample size to be 300 and the number of cues which is the fraction of the sample is 250 while the sample proportion is 250 divided by 300 and that gives us 0 .83 so we are trying to test the claim that the proportion of q's must not exceed 0 .8 that is the highest proportion of q's that we can have is 0 .8 so that simply implies that h not is that p is lesser than or equals to 0 .8 like that is the highest proportion we can have but as the statistician we are going against the sentence that's no there's a chance that it can actually be higher so our alternative hypothesis h1 is going to be p excuse me is greater than 0 .8 now we'll be needing a decision rule and it has been stated that our level of significance is 0 .04 so very important write it out alpha is equal to 0 .04 so to make our decision i will be making use of the p value method and the rule making each of the p value method is that if your p value is greater than alpha we are going to accept our null hypothesis and if your p value is going to be lesser than alpha we are going to reject the null hypothesis so to get a p value we need to get our z score first so it is from the z score we're able to get a p value so our z score is equals to p cap minus p divided by the square root of p times q divided by n and our q in this case is one minus p which gives us 0 .8 and sorry our p is equal to our q rather is a 1 minus p that has 1 minus 0 .8 which gives us 0 .2 hence we can substitute the whole of the parameters into the formula we have z is equal to 0 .83 minus 3 minus 0 .8 divided by the square root of 0 .8 times 0 .2 divided by 300 so 0 .83 minus 0 .03 we have 0 .03 3 divided by so the square root of 0 .8 times 0 .3 divided by 300 for us to have 0 .02 8 so so when we do the math, that is 0 .03 divided by 0 .0 to 8 .2, we have our z score to be 1 .06.

03:22 So let us try to visualize what does z score simply means.

03:26 So we have something of this nature...

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