Question

The following n = 10 observations are a sample from a normal population. 7.4 7.0 6.5 7.4 7.6 6.2 6.9 7.6 6.5 6.9 (a) Find the mean and standard deviation of these data. (Round your standard deviation to four decimal places.) mean standard deviation (b) Test H0: μ = 7.5 versus Ha: μ < 7.5. Use α = 0.01. Test statistic: t = __.

Answer 1

Mean, x: 7

standard deviation(s)=0.4944

t=-3.1980

Explanation:

given data

7.4 7.0 6.5 7.4 7.6 6.2 6.9 7.6 6.5 6.9

Count, N: 10

Sum, Σx: 70

Mean, x: 7

Variance, s2: 0.24444444444444

Steps

s2 =
`(∑(xi - x)2)/(N - 1)`

=
`((7.4 - 7)2 + ... + (6.9 - 7)2)/(10-1)`

=
`(2.2)/(9)`

=0.24444444444444

s =
`√(0.24444444444444)`

⇒s= 0.49441323247304

⇒s=0.4944

b) H0: μ = 7.5 versus Ha: μ < 7.5.

α = 0.01.

t=
`(x-u)/((s)/(√(n) ) )`

t=
`(7-7.5)/((0.4944)/(√(10) ) )`

t=-3.1980

for 9 df and α = 0.01

t stat from table is -2.82

as t test is less than t

we reject the null hypothesis