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Problem 49 Hard Difficulty

The proportion of individuals insured by the All-Driver Automobile Insurance Company who received at least one traffic ticket during a five-year period is. 15.
a. Show the sampling distribution of $\overline{p}$ if a random sample of 150 insured individuals is ultimate the proportion having received at least one ticket.
b. What is the probability that the sample proportion will be within $\pm .03$ of the population proportion?


a. See graph
b. 0.6970


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Video Transcript

all right, We're given that the proportion of individuals who are insured under a certain insurance company get at least one traffic ticket, uh, is 0.15 15%. So for part A, it wants us to draw a sampling distribution graph for an equals 150. So first off, let's check if we can use a normal distribution to approximate this. So on times P you check with that? What's first check end times P, which is 22.5, which is created them five. So we're good. And then in Thames, one minus p, that's 1 50 times 500.85 which is 127.5, which is also definitely greater than five. So we are good. Thio use a normal distribution. So let's just get this out. Yeah, nothing. Nicest looking normal curve. But it's close enough, I guess. Anyway, uh, first off, let's find her mean creepy, huh? Not be happy bar. Some other sources use, uh, he had, but that's besides the point. That's just gonna eagle population to that 0.15 Standard deviation. P bar just p. Times one minus. Whoa! Hello. That was weird. one minus p all over And take the square root of it. Uh, when you multiply that out. Sorry about that. When you multiply that out, you get approximately 0.29 Okay, knowing this, let's get a sketch of autograph. Looks like. All right, so this mean it's gonna be 0.15 and then, Ah, this could be 0.179 and 0.1 to one. Yes, I believe just adding that mentally me had Yes. So And this will be this is one standard deviation away. That's terribly tiny. Oops. Ah, good enough. And then it's also one. There we go. That looks significantly neater. All right. For part B, I want to find the probability that whatever sample proportion we find is within 0.3 of our population proportions. So you need R Z lower and the upper. So we have a negative 0.3 over our standard deviation. 0.29 and 0.3 over 0.29 When you calculate those out, you get 1.3 negative. 1.3 If we look at our ze table. The Z corresponds to 0.1515 This one corresponds to open a patient up. Easy, easy. Should be peas. Ah, that. Okay, My eraser tools having issues today. So, uh, another dots. Oh, there we go. Those there we go anyway. This should be a P lower the SVP upper a 0.8485 So a probability is gonna be p upper minus p lower, which gives us 0.6970 Nice. And there we have it.