A common characterization of obese individuals is that their body mass index is

Question

A common characterization of obese individuals is that their body mass index is at least 30 BMI = weight/ height)2 where height is meters and weight is in kilograms . An article reported that in a sample of female workers, 264 had BMIs of less than 25, 156 had BMIs that were at least 25 but less than 30, and 122 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 Ho: p = 0.20 Ha: p 0.20 Ho: > 0.20 Hai p = 0.20 Ho: p = 0.20 Ha: p> 0.20 Ho: = 0.20 Hai 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.) p-value =

Explanation / Answer

Solution:-

n = 542

x = 122

p = 0.2251

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P < 0.20

Alternative hypothesis: P > 0.20

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation () and compute the z-score test statistic (z).

= sqrt[ P * ( 1 - P ) / n ]

= 0.0172

z = (p - P) /

z = 1.46

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is more than 1.46. We use the Normal Distribution Calculator to find P(z > 1.46).

Thus, the P-value = 0.0721

Interpret results. Since the P-value (0.0721) is more than the significance level (0.05), we have to accept the null hypothesis.

The probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 24%

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