Question

Normal (or average) body temperature of humans is often thought to be 98.6 ∘

F. Is that number really the average? To test this, we will use a data set obtained from 67 healthy female volunteers aged 18 to 40 that were participating in vaccine trials. We will assume this sample is representative of a population of all healthy females. The mean body temperature for the 67 females in our sample is 98.85 ∘
F and the standard deviation is 0.852 ∘
F. The data are not strongly skewed. Use the Theory-Based Inference applet to find a 95% confidence interval for the population mean body temperature for healthy females. Round your answers to 2 decimal places, e.g. 0.75. The confidence interval is to eTextbook and Media Based on your confidence interval, is 98.6 ∘
F a plausible value for the population average body temperature or is the average significantly more or less than 98.6 ∘
F ? Explain how you are determining this. eTextbook and Media In the context of this study, was it valid to use the theory-based (t-distribution) approach to find a confidence interval? Yes No eTextbook and Media

Answer 1

Step-by-step explanation:

To find a 95% confidence interval for the population mean body temperature of healthy females, we can use the theory-based (t-distribution) approach since the sample size is relatively small (n = 67) and the population standard deviation is unknown.

Using the information provided, the mean body temperature for the sample is 98.85 °F, and the standard deviation is 0.852 °F. The formula for the confidence interval is:

Confidence Interval = sample mean ± (t-value * standard error)

First, we need to determine the t-value for a 95% confidence level with (n-1) degrees of freedom. In this case, the degrees of freedom are 67 - 1 = 66. Using a t-table or statistical software, the t-value for a 95% confidence level and 66 degrees of freedom is approximately 1.997.

Next, we calculate the standard error, which is the standard deviation divided by the square root of the sample size:

Standard Error = standard deviation / √sample size

= 0.852 / √67

≈ 0.104

Now we can calculate the confidence interval:

Confidence Interval = 98.85 ± (1.997 * 0.104)

Simplifying the calculation:

Confidence Interval ≈ 98.85 ± 0.208

Rounding to two decimal places, the 95% confidence interval for the population mean body temperature is approximately (98.64, 99.06) °F.

To determine whether 98.6 °F is a plausible value for the population average body temperature, we need to check if it falls within the confidence interval. In this case, since 98.6 °F is within the confidence interval of (98.64, 99.06) °F, it is a plausible value for the population average body temperature. This means that based on the sample data, there is no strong evidence to suggest that the average body temperature significantly differs from 98.6 °F.

In terms of using the theory-based (t-distribution) approach to find the confidence interval, it is valid in this study because the sample size is small and the population standard deviation is unknown. The t-distribution accounts for the uncertainty introduced by using the sample standard deviation to estimate the population standard deviation.