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

Welcome 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 halter negative hypothesis. So basically, what's happening here is that we think the proportion, 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 h o is that p, that's a proportion, is equal to 0.30. Now the alternative hypothesis, things you're trying to find evidence for, that's h a, is that p is actually is greater than 0.30. To do this, you were 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, it's definitely not gonna be the variance one. So it's basically just gonna 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 gonna 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 some portion 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 0.30. 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 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 m. 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 0.38-0.30 divided by the square roots. This is gonna be 0.30 and then 1 minus 0.30 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 wanna do this, just be very careful how you plug this into your calculator. Alright? So what you're gonna 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 0 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 0.30. And what this test statistic over here tells you is basically how far away you are from that actual, mean of 0.30. Remember, we're trying to figure out or we're trying to find evidence for p is actually greater than 0.30. And what we actually found over here is that with our sample statistic where p hats 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.

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

Statistics

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

Struggling with Statistics?

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Multiple Choice

A survey claimed that $30 percent sign$ of adults prefer electric cars over traditional cars. A car manufacturer believes the true proportion is higher than $30 percent sign$ . 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.

$z equals 1.23$

$z = - 1.23$

$z equals 4.38$

$z = - 4.38$

Verified step by step guidance

Identify the type of hypothesis test needed. Since we are dealing with proportions, we will use the z-test for proportions.

Define the null hypothesis (H0) and the alternative hypothesis (H1). Here, H0: p = 0.30 and H1: p > 0.30, where p is the true proportion of adults who prefer electric cars.

Calculate the sample proportion (p̂) using the formula p̂ = x/n, where x is the number of successes (19 adults) and n is the sample size (50 adults).

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

Substitute the values into the formula and simplify to find the z-test statistic. This will help determine if the sample proportion is significantly higher than the hypothesized proportion.

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Multiple Choice

A null hypothesis can only be rejected at the $5 %$ significance level if and only if which of the following is true?

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Textbook Question

"[DATA] Height versus Head Circumference [See Problem 13 in Section 12.3] A pediatrician wants to determine the relation that may exist between a child’s height and head circumference. She randomly selects 11 children from her practice, measures their heights and head circumferences, and obtains the following data:

b. Suppose the researcher wanted to determine if there is a positive association between height and head circumference. What would be the null and alternative hypotheses?"

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Multiple Choice

A popular theme park claims that their weekly attendance is around $100 comma 000$ . You believe that the weekly attendance is different than this claimed value, so you gather sample data. Write the null and alternative hypotheses.

A candy manufacturer seeking to minimize the variation in weights of their candies claims to produce candies with a standard deviation less than $0.300$ g. Write the null and alternative hypotheses.

Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.

$H sub 0 colon$ $p equals 0.4$

$H_{a} :$ $p is not equal to 0.4$

In a certain hypothesis test, $H sub 0 colon p equals 0.4$ , $H_{a} : p$ < $0.4$ . You collect a sample and calculate a test statistic $z = - 1.32$ . Find the $P$ -value.

Stem Cell Survey In a Newsweek poll of 882 adults, 481 (or 55%) said that they were in favor of using federal tax money to fund medical research using stem cells obtained from human embryos. A politician claims that people don’t really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Use the following probabilities related to determining whether the result of 481 is significantly high (assuming the true rate is 50%). Is 481 significantly high? What should be concluded about the politician’s claim? Explain.

P(respondent says to use the federal tax money) = 0.5

P(among 882, exactly 481 says to use federal tax money) = 0.000713

P(among 882,481 or more say to use federal tax money) = 0.00389

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Textbook Question

Final Conclusions

In Exercises 21–24, use a significance level of α = 0.05 and use the given information for the following:

State a conclusion about the null hypothesis. (Reject H0 or fail to reject H0.)

Without using technical terms or symbols, state a final conclusion that addresses the original claim.

Original claim: The mean pulse rate (in beats per minute) of adult males is 72 bpm. The hypothesis test results in a P-value of 0.0095.

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