Question

A common characterization of obese individuals is that their body mass index is at least 30 [BMI = weight/(height)2, where height is in meters and weight is in kilograms]. An article reported that in a sample of female workers, 264 had BMIs of less than 25, 157 had BMIs that were at least 25 but less than 30, and 121 had BMIs exceeding 30. Is there compelling evidence for concluding that more than 20% of the individuals in the sampled population are obese? (a) State the appropriate hypotheses with a significance level of 0.05. H0: p = 0.20 Ha: p ≠ 0.20H0: p > 0.20 Ha: p = 0.20 H0: p = 0.20 Ha: p < 0.20H0: p = 0.20 Ha: p > 0.20 Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = _. P-Value = _. What can you conclude? Reject the null hypothesis. There is not sufficient evidence that more than 20% of the population of female workers is obese.Do not reject the null hypothesis. There is sufficient evidence that more than 20% of the population of female workers is obese. Reject the null hypothesis. There is sufficient evidence that more than 20% of the population of female workers is obese.Do not reject the null hypothesis. There is not sufficient evidence that more than 20% of the population of female workers is obese. (b) Explain in the context of this scenario what constitutes type I error. A type I error would be declaring that 20% or less of the population of female workers is obese, when in fact more than 20% are actually obese.A type I error would be declaring that 20% or more of the population of female workers is obese, when in fact less than 20% are actually obese. A type I error would be declaring that less than 20% of the population of female workers is obese, when in fact 20% or more are actually obese.A type I error would be declaring that more than 20% of the population of female workers is obese, when in fact 20% or less are actually obese. Explain in the context of this scenario what constitutes type II error. A type II error would be declaring that 20% or less of the population of female workers is obese, when in fact more than 20% are actually obese.A type II error would be declaring that 20% or more of the population of female workers is obese, when in fact less than 20% are actually obese. A type II error would be declaring that less than 20% of the population of female workers is obese, when in fact 20% or more are actually obese.A type II error would be declaring that more than 20% of the population of female workers is obese, when in fact 20% or less are actually obese. (c) What is the probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 26%? (Round your answer to four decimal places.)

Name: a common characterization of obese individuals is that their body mass index is at least 30 bmi weightheight2 where height is in meters and weight is in kilograms an article reported that in 03296
Uploaded: 2022-11-23T18:17:35-08:00
Duration: 14 min 41 s
Channel: Jon Southam
Description: a common characterization of obese individuals is that their body mass index is at least 30 bmi weightheight2 where height is in meters and weight is in kilograms an article reported that in 03296

          A common characterization of obese individuals is that their
body mass index is at least 30 [BMI =
weight/(height)2, where height is in meters and
weight is in kilograms]. An article reported that in a sample of
female workers, 264 had BMIs of less than
25, 157 had BMIs that were at least 25 but less than 30,
and 121 had BMIs exceeding 30. Is there compelling
evidence for concluding that more than 20% of the individuals in
the sampled population are obese?
(a) State the appropriate hypotheses with a significance level
of 0.05.
H0: p = 0.20
Ha: p ≠ 0.20H0: p >
0.20
Ha: p =
0.20    H0: p =
0.20
Ha: p <
0.20H0: p = 0.20
Ha: p > 0.20
Calculate the test statistic and determine
the P-value. (Round your test statistic to two
decimal places and your P-value to four decimal
places.)
z = _______.
P-Value = _______.
What can you conclude?
Reject the null hypothesis. There is not sufficient evidence
that more than 20% of the population of female workers is obese.Do
not reject the null hypothesis. There is sufficient evidence that
more than 20% of the population of female workers is
obese.    Reject the null hypothesis. There is
sufficient evidence that more than 20% of the population of female
workers is obese.Do not reject the null hypothesis. There is not
sufficient evidence that more than 20% of the population of female
workers is obese.
(b) Explain in the context of this scenario what constitutes
type I error.
A type I error would be declaring that 20% or less of the
population of female workers is obese, when in fact more than 20%
are actually obese.A type I error would be declaring that 20% or
more of the population of female workers is obese, when in fact
less than 20% are actually obese.    A type I
error would be declaring that less than 20% of the population of
female workers is obese, when in fact 20% or more are actually
obese.A type I error would be declaring that more than 20% of the
population of female workers is obese, when in fact 20% or less are
actually obese.
Explain in the context of this scenario what constitutes type II
error.
A type II error would be declaring that 20% or less of the
population of female workers is obese, when in fact more than 20%
are actually obese.A type II error would be declaring that 20% or
more of the population of female workers is obese, when in fact
less than 20% are actually obese.    A type II
error would be declaring that less than 20% of the population of
female workers is obese, when in fact 20% or more are actually
obese.A type II error would be declaring that more than 20% of the
population of female workers is obese, when in fact 20% or less are
actually obese.
(c) What is the probability of not concluding that more than 20%
of the population is obese when the actual percentage of obese
individuals is 26%? (Round your answer to four decimal
places.)

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Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Jon Southam

11/23/2022

Step 1

20 (Null hypothesis: 20% of the population is obese) Ha: p > 0.20 (Alternative hypothesis: More than 20% of the population is obese) The total number of individuals in the sample is 264 + 157 + 121 = 542. The number of obese individuals (BMI > 30) is 121. So, the Show more…

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A common characterization of obese individuals is that their body mass index is at least 30 [BMI = weight/(height)2, where height is in meters and weight is in kilograms]. An article reported that in a sample of female workers, 264 had BMIs of less than 25, 157 had BMIs that were at least 25 but less than 30, and 121 had BMIs exceeding 30. Is there compelling evidence for concluding that more than 20% of the individuals in the sampled population are obese? (a) State the appropriate hypotheses with a significance level of 0.05. H0: p = 0.20 Ha: p ≠ 0.20H0: p > 0.20 Ha: p = 0.20 H0: p = 0.20 Ha: p < 0.20H0: p = 0.20 Ha: p > 0.20 Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = _______. P-Value = _______. What can you conclude? Reject the null hypothesis. There is not sufficient evidence that more than 20% of the population of female workers is obese.Do not reject the null hypothesis. There is sufficient evidence that more than 20% of the population of female workers is obese. Reject the null hypothesis. There is sufficient evidence that more than 20% of the population of female workers is obese.Do not reject the null hypothesis. There is not sufficient evidence that more than 20% of the population of female workers is obese. (b) Explain in the context of this scenario what constitutes type I error. A type I error would be declaring that 20% or less of the population of female workers is obese, when in fact more than 20% are actually obese.A type I error would be declaring that 20% or more of the population of female workers is obese, when in fact less than 20% are actually obese. A type I error would be declaring that less than 20% of the population of female workers is obese, when in fact 20% or more are actually obese.A type I error would be declaring that more than 20% of the population of female workers is obese, when in fact 20% or less are actually obese. Explain in the context of this scenario what constitutes type II error. A type II error would be declaring that 20% or less of the population of female workers is obese, when in fact more than 20% are actually obese.A type II error would be declaring that 20% or more of the population of female workers is obese, when in fact less than 20% are actually obese. A type II error would be declaring that less than 20% of the population of female workers is obese, when in fact 20% or more are actually obese.A type II error would be declaring that more than 20% of the population of female workers is obese, when in fact 20% or less are actually obese. (c) What is the probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 26%? (Round your answer to four decimal places.)

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Transcript

00:01 So we're looking at body mass index.

00:06 Because that can be a way to determine obesity.

00:12 And the rule of the figure is that if the bmi is at least 30, then the person is going to be considered obese.

00:22 So bmi greater than 30, greater than equal to 30, then we're obese.

00:32 So we're given some data.

00:35 Were told a sample of female workers and what their bmi's were.

00:44 And the only one we care about is the numbers that have bmi's exceeding 30.

00:51 And of the sample of 542, there were a total of 121 that had bmi's at least 30, bmi greater than equal to 30.

01:06 And we're given some other data.

01:09 We're given like 264 had bmi's less than 25, less than 25.

01:22 And then we had 157 that were between 25 and 30.

01:32 Bmi was between 30 and 25, including 25.

01:40 And then these 121 were 30 or above.

01:43 So like i said, we don't really care about these ones in terms of our study because we are testing.

01:50 The claim that more than 20 % of individuals of this population are obese.

01:56 So our null hypothesis, let me get rid of this because we don't need it anymore.

02:03 Our null hypothesis is that the proportion of the population that was obese is 0 .2.

02:11 And the alternative hypothesis is that it exceeds that value.

02:16 It's greater than 0 .2.

02:22 All right, so this is all for part a.

02:27 So this is an alpha part a here.

02:29 And we're also going to calculate the z statistic and the p value here.

02:35 So let's get a picture of this.

02:36 So here's where the p value is 0 .2.

02:40 And we're told to test at the alpha of 0 .05.

02:45 And what that means is we have some p critical value here, such that the area to the right of it is 0 .05.

02:56 And this is that critical region that if we fall within this, that means we're significantly above point two to say that we could reject the null hypothesis and say we're greater than.

03:06 So our z statistic, our one proportion z statistic, is calculated as p -not, which is our proportion of observed people with bmi is greater than 30, minus the assume population proportion divided by the square root of p times one minus p all over n, and ends the total number in the sample.

03:30 So p0 is 121 over 542.

03:47 And that ends up being about 0 .223.

03:56 But for a purpose, i usually don't round to the very end.

03:59 So it can be a little more accurate as i'm going through.

04:02 So i'll just leave it as 121 over 542, minus 0 .2 all over the square root of, 0 .2 times 1 minus 0 .2 all over 542.

04:21 And this gives us a z score of 1 .35.

04:34 And the p value associated with this.

04:37 So we want the probability of z being greater than this value, which is going to fall somewhere up here.

04:49 And the way we do that is you do 1 .5.

04:53 The probability that z is less than 1 .35.

04:57 And you could use a table or a spreadsheet.

05:00 And i use my spreadsheet to get the p value.

05:04 And it's 0 .912.

05:09 Oh, we go to four decimal places.

05:11 So it's good, this z score.

05:14 And i use my spreadsheet, like i said.

05:16 So i used 1 .353 .0.

05:26 I put that in into my spreadsheet.

05:31 And i use this function called norm s.

05:33 Dist.

05:42 And what you do is you put in your z score and then out pops the air to the left.

05:48 And we end up with 0 .9.

05:59 Yeah, 0 .912 is going round.

06:01 We get that.

06:04 Which gives us a p value of 0 .088.

06:20 So because that p value is not less than the alpha, we fail to reject.

06:25 And we'd say that the this is while this is a proportion that's greater than 0 .2, we can assume it's likely to a chance and this result is not significant though.

06:42 So the p value, the proportion is still 0 .2...

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