In: Math
You decide to further your research project by hypothesizing that the true proportion of core body temperature increases amidst higher ambient temperature and humidity levels for the population who do not use electric fans is less than those who do use electric fans, setting the level of significance at 5% for the formal hypothesis test. In other words, you extend your sampling to two samples instead of just one. You randomly sample 31 and 39 participants for your first and second groups, respectively, based on your research funding and for 45 minutes, all study participants sit in a chamber maintained at a temperature of 108 degrees Fahrenheit (i.e., 42 degrees Celsius) and a relative humidity of 70%. After the first 45-minute warming period, you record the participants' core body temperatures. Furthermore, for Group 2 only you place a personal sized electric fan 3 feet away with its airflow directed at a given participant's chest area, and the participants relax in this position for the next 45 minutes, whereas for Group 1 you do not provide electric fans. At the end of this 45-minute fan period, you record the core body temperatures of all participants, documenting any temperature increases as compared to the start of the time period. The following table comprises the data you collect.
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Per Step 3 of the 5-Steps to Hypothesis Testing, choose the appropriate decision rule.
Please note that 0 and 1 are defined as no and yes, respectively, which is a typical coding scheme in Public Health.
Select one:
a. Accept H0 if t < -1.282
b. Reject H0 if z ≤ -1.645
c. Reject H0 if t = +1.282
d. Reject H1 if z ≥ -2.326
**P.S: can you please explain how to get Z, and when to make it negative, i.e. why we choose upper tail and then make it - ? . Thanks alot
Solution:-
p1 = 10/31 = 0.3226
p2 = 15/39 = 0.3846
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P1> P2
Alternative hypothesis: PNo Fan < PFan
Note that these hypotheses constitute a one-tailed test.
Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method is a two-proportion z-test.
Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).
p = (p1 * n1 + p2 * n2) / (n1 + n2)
p = 0.35714
SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2)
] }
SE = 0.1153
z = (p1 - p2) / SE
z = - 0.538
zcritical = - 1.645
b) Reject H0 if z ≤ -1.645
where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.
Interpret results. Since the z-value (- 0.538) is does not lies in rejection region, hence we cannot reject the null hypothesis.
From the above test we do not have sufficient evidence evidence in the favor of the claim that the true proportion of core body temperature increases amidst higher ambient temperature and humidity levels for the population who do not use electric fans is less than those who do use electric fans, setting the level of significance at 5% for the formal hypothesis test.
Since there is less than sign in the alternate hypothesis, hence this test would be left tailed test, so we will find out the zcritical value for the 0.05 p-value in the left tail, that would be -ve z-value.