Question

One year, the mean age of an inmate on death row was 38.8 years. A sociologist wondered whether the mean age of a death-row inmate has changed since then. She randomly selects 32 death-row inmates and finds that their mean age is 37.5, with a standard deviation of 9.2. Construct a 95% confidence interval about the mean age. What does the interval imply

Answer 1

The confidence interval about the given mean age is (40.7, 34.3). Since the given mean age of 38.8 years is in the obtained interval, there is no sufficient evidence to conclude that the mean age had changed.

What is the formula for calculating the confidence interval?

The formula for the confidence interval is

C.I = $\bar x$ ± z(σ/√n)

Where $\bar x$ - sample mean; z or z(α/2) - test value; σ - standard deviation of the sample; n - sample size;

Calculation:

Consider the hypothesis,

null hypothesis H0: μ = 38.8

alternate hypothesis Ha: μ ≠ 38.8

It is given that,

sample size n = 32

sample mean $\bar x$ = 37.5

standard deviation σ = 9.2

Constructing a 95% confidence interval about the mean age:

a) Finding the significance level:

= 1 - (95/100)

= 1 - 0.95

∴ α = 0.05

b) Finding the z-value for the obtained significance level:

z(α/2) = z(0.05/2)

∴ z = 1.96 (From the distribution table)

c) Calculating the lower and upper bounds:

C.I = $\bar x$  ± z(σ/√n)

On substituting,

C.I = 37.5 ± (1.96)(9.2/√32)

⇒ 37.5 ± (1.96)(1.626)

⇒ 37.5 ± 3.186(≅3.2)

Upper bound = 37.5 + 3.2 = 40.7

Lower bound = 37.5 - 3.2 = 34.3

Thus, the interval is from 34.3 years to 40.7 years.

Hence the given mean age of 38.8 is included in the interval, the evidence is insufficient to conclude that there is a change in the mean age.

Learn more about the confidence interval here:

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