1. A set of researchers believed that the normal body temperature of children is

Question

1.       A set of researchers believed that the normal body temperature of children is actually more than 98.6 degrees. They measured the temperature in 14 randomly selected healthy adolescents and found a sample mean of 98.9 with a standard deviation of 0.7 degrees with the temperatures being normally distributed.

a.       Write the null and alternate hypothesis.

b.       Calculate the test statistic and write a conclusion for this question.

c.       Now suppose a sample of 125 adolescents was taken and the same mean (98.9) and standard deviation (.7) was achieved. Repeat the test.

d.       Explain what caused the difference between the outcomes for parts b and c.

Explanation / Answer

Solution

Let X = body temperature, be Normally distributed with mean µ and variance 2.

Part (a)

Null hypothesis H0: µ = 98.6 Vs alternative HA: µ > 98.6 ANSWER

Part (b) [level of significance is taken to be 5%]

Test Statistic: t = (n){(X bar – 98.6)/s} has a t-distribution with (n - 1) degrees of freedom, where s = sample standard deviation and n = sample size.

= (14){(98.9 – 98.6)/0.7} = (3.7417 x 0.3)/0.7 = 1.604 ANSWER 1

Being a greater-than-type one-sided test, the critical value is upper 5% point of t-distribution with 13 degrees of freedom, which is read from Standard Statistical Tables, as 1.771

Since calculated value, 1.604 is less than the table value, 1.771, H0 is accepted.

=> there is no evidence to suggest that average body temperature is greater than 98.6 ANSWER2

Part (c)

Null hypothesis H0: µ = 98.6 Vs alternative HA: µ > 98.6

[level of significance is taken to be 5%]

Test Statistic: t = (n){(X bar – 98.6)/s} has a t-distribution with (n - 1) degrees of freedom, where s = sample standard deviation and n = sample size.

= (125){(98.9 – 98.6)/0.7} = (11.1803 x 0.3)/0.7 = 4.792

Being a greater-than-type one-sided test, the critical value is upper 5% point of t-distribution with 124 degrees of freedom, which is read from Standard Statistical Tables, as lying between 1.660 and 1.645.

Since calculated value, 4.792 is greater than the table value, H0 is rejcted.

=> there is evidence to suggest that average body temperature is greater than 98.6 ANSWER

Part (d)

As a general rule, as sample size increases, the reliability of estimates of population parameters based on sample values improves considerably.

Thus, when the sample size increased considerably, 9 times from 14 to 125, the observed difference of 0.3 became significant. ANSWER

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