+ Question Details My Notes Ask Your Teacher Body temperatures of humans (in deg

Question

+ Question Details My Notes Ask Your Teacher Body temperatures of humans (in degrees Fahrenheit) have a known standard deviation of 0.80 degree. A random sample of 24 people yielded a mean of x-98.6 degrees with a sample standard deviation of s= 0.72 degrees. It is known that the human body temperatures has a normal distribution, we want to estimate the true average human body temperature, , (in degrees Fahrenheit) a)What is the critical value for a 96% confidence interval for b) Create a 96% confidence interval for c) How many observations would we need to guarantee that the 96% confidence interval has has a length of 0.2 or less? d) Create a 96% prediction interval for the body temperature of a single human. e) Assuming is not known, create a 96% confidence interval for using this data

Explanation / Answer

(a) Here critical value for 96% confidence interval for = 2.054

(b) 96% confidence intervla = x +- Z96% (/n)

n = 24

= 0.80 C

= 98.6 +- 2.05375 * (0.80/ 24)

= 98.6 +- 2.05375 * 0.1633

= (98.265, 98.935)

(c) Let say n is requisite sampel size

so the length of confidence interval < 0.2

so margin of error < 0.1

margin of error = critical test statistic * standard error of the mean

2.05375 * (0.80/ n) < 0.1

n > 2.05375 * 0.8/0.1

n > 269.94

or n would be 270

(d) 96% prediction interval for a single human = x +- Z96%

= 98.6 +- 2.05375 * (0.80)

= (96.957, 100.243)

(e) 96% confidence interval when is not known.

96% confidence interval = x +- t23,0.04 (s/n)

Here dF= 24-1 = 23 and alpha = 0.04

so tcritical = t23,0.04 = 2.177

96% confidence interval = x +- t23,0.04 (s/n) = 98.6 +- 2.177 * (0.72/ 24)

= 98.6 +- 0.32

= (98.28, 98.92)

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