Question

Researchers studying the human voice measure the volume in decibels. Normal conversation occurs at around

50 decibels but some singers can reach over 100 decibels. The researchers choose a random sample of 25

professional opera singers and record the maximum volume reached by each. They find the mean volume for

this sample is 98 decibels with a standard deviation of 10 decibels. Carry out a hypothesis test with alpha= 0.05,

to determine if the mean maximum volume of all professional opera singers is 100 decibels.

Answer 1

Solution:

Here, we have to use one sample t test for the population mean.

The null and alternative hypotheses are given as below:

Null hypothesis: H0: the mean maximum volume of all professional opera singers is 100 decibels.

Alternative hypothesis: Ha: the mean maximum volume of all professional opera singers is not 100 decibels.

H0: µ = 100 versus Ha: µ ≠ 100

This is a two tailed test.

The test statistic formula is given as below:

t = (Xbar - µ)/[S/sqrt(n)]

From given data, we have

µ = 100

Xbar = 98

S = 10

n = 25

df = n – 1 = 24

α = 0.05

Critical value = - 2.0639 and 2.0639

(by using t-table or excel)

t = (Xbar - µ)/[S/sqrt(n)]

t = (98 – 100)/[10/sqrt(25)]

t = -2/2

t = -1.00

P-value = 0.3273

(by using t-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that the mean maximum volume of all professional opera singers is 100 decibels.