A survey claimed that 30 % 30\% 30% of adults prefer electric cars over traditional cars . A car manufacturer believes the true proportion is higher than 30 % 30\% 30% . To test this, they survey a random sample of 50 50 50 adults and find that 19 19 19 say they prefer electric cars . Determine which test statistic to use & calculate it .

back, everyone. So now that we understand how to calculate test statistics, let's take a look at this practice problem. So we have a survey that claims that 30% of adults prefer electric cars over traditional cars. Now a car manufacturer has reason to believe that that proportion is actually higher than 30%. So in order to test this claim, they survey a random sample of 50 adults, and they find that 19 of them say that they prefer electric cars. So we're gonna determine which test statistic to use, and then we're gonna go ahead and calculate it. Alright? Now remember, even though this is about test statistics, a good thing to always do is just write out the null and alternative hypothesis. So basically, what's happening here is that we think the proportion, is the claim about the population is that 30% of adults prefer electric cars. In other words, that is gonna be your null hypothesis. So your HO is that p, that's a proportion, is equal to 0.3. Now the alternative hypothesis, things you're trying to find evidence for, that's h a, is that p is actually is greater than 0.3. To do this, you you get that sample. Right? So basically, this null and alternative hypothesis is always really good thing to write out because it's gonna tell you the value that you're supposed to use. Alright? So now let's go ahead and figure out which test statistic you're gonna use from your sample. Remember, it really all comes down to, and you've already done all of this before, which one of the parameters are you actually looking at here? So we have means, proportions, and variance. In this case, we actually have from our null hypothesis, we're looking at proportions here. So it's not gonna be one of these equations, and it's definitely not going to be the variance one. So that's basically just going to be our z score for proportions, which, again, we've already seen how to do this. Alright? So really, there's not really a whole lot that goes into selecting which type of equation, when it comes to proportions because you're only just going to use one of them. Alright? So basically, what's gonna happen over here is that we have that z is equal to and I'm just gonna rewrite this. We have the p hats. Remember, p hat is the sample proportion minus regular p, that's the population proportion, just unfortunate that all these things are p's, divided by the square root of p one minus p divided by the square root of n. Alright? So again, we just have to figure out what all of these things are over here. So we actually already know what value to use for p because remember, we have p is equal to 0.3. We're assuming that to be true. So we have this p, this p, and this p. We already have what n is equal to. That's just the sample that we used in this case, which was 50. So all we have to do really is just figure out what this p hat is. What's the sample proportion? We randomly surveyed 50 adults. 19 of them said that they prefer electric cars. That's from our sample. So, basically, what we can do over here is is we're gonna have to figure out, well, first, what is p hat? Alright? And I'm actually gonna do this, over here. So p hat, remember, is just equal to x over n, where x is just the number of successes that you have. In this case, the success is really just that 19 saying that they prefer electric electric cars. So in this case, this is your x and this is your n. So this is basically just 19 divided by 50. And if you work this out, you're gonna get 0.38. Alright. So this is a little intermediate step that you might have to do before you actually plug everything into your equation. So we have that Z is equal to, then now we have what p hat is equal to. This is just going to be 0.38 minuteus 0.3 divided by the square roots. This is going to be 0.3 and then one minus 0.3 and then divided by the square roots of 50 over here. Alright. So again, you're in this long calculation. Go ahead and just make sure that you're doing it correctly. You could do the denominator first and then do the, do the division for the numerator. However you want to do this. Just be very careful how you plug this into your calculator. All right. So what you're going to see if you've done this correctly is that your Z score should be equal to once you're done. It should be equal to zero oh, sorry. 1.23. Alright? So this is gonna be your test statistic. Again, this 1.23 is gonna tell you basically how far away you are from the mean. You're assuming that this proportion over here, this this, this normal distribution, is gonna have a peak at 0.3. And what this test statistic over here tells you is basically how far away you are from that actual, mean of 0.3. Remember, we're trying to figure out or we're trying to find evidence for p is actually greater than 0.3. And what we actually found over here is that with our sample statistic where p hat was equal to 0.38, that is a z score of 1.23 to the right. So in other words, that's basically over here. Alright? So that's our test statistic, the z equals 1.23. Let me know if you have any questions. Thanks for watching. Hopefully, you got the right answer.

A survey claimed that 30 % 30\% 30% of adults prefer electric cars over traditional cars . A car manufacturer believes the true proportion is higher than 30 % 30\% 30% . To test this, they survey a random sample of 50 50 50 adults and find that 19 19 19 say they prefer electric cars . Determine which test statistic to use & calculate it .

z = − 1.23 z=-1.23 z = − 1.23

z = − 4.38 z=-4.38 z = − 4.38

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

Statistics for Business

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

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Multiple Choice

A survey claimed that $30\%$ of adults prefer electric cars over traditional cars. A car manufacturer believes the true proportion is higher than $30\%$ . To test this, they survey a random sample of $50$ adults and find that $19$ say they prefer electric cars. Determine which test statistic to use & calculate it.

$z=1.23$

$z=-1.23$

$z=4.38$

$z=-4.38$

Verified step by step guidance

Identify the type of test statistic needed: Since the problem involves testing a proportion, we will use the formula for the z-test for proportions.

Define the null hypothesis (H0) and the alternative hypothesis (H1): H0: p = 0.30 (the proportion of adults preferring electric cars is 30%), H1: p > 0.30 (the proportion is greater than 30%).

Calculate the sample proportion (p̂): p̂ = x/n, where x is the number of adults preferring electric cars (19) and n is the sample size (50).

Use the z-test formula for proportions: z = (p̂ - p) / sqrt(p(1-p)/n), where p is the hypothesized proportion (0.30), p̂ is the sample proportion, and n is the sample size.

Substitute the values into the formula and simplify: Plug in p̂, p, and n into the formula to find the z-value, which will help determine if the null hypothesis can be rejected.

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