Question

A)

You wish to test the following claim (HaHa) at a significance level of α=0.01α=0.01.

      Ho:μ=62.3Ho:μ=62.3
      Ha:μ<62.3Ha:μ<62.3

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=36n=36 with a mean of M=60.1M=60.1 and a standard deviation of SD=14.5SD=14.5.

What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value =

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

The test statistic is...

• in the critical region
• not in the critical region

This test statistic leads to a decision to...

• reject the null
• accept the null
• fail to reject the null

As such, the final conclusion is that...

• There is sufficient evidence to warrant rejection of the claim that the population mean is less than 62.3.
• There is not sufficient evidence to warrant rejection of the claim that the population mean is less than 62.3.
• The sample data support the claim that the population mean is less than 62.3.
• There is not sufficient sample evidence to support the claim that the population mean is less than 62.3.

B)

The distributions with large n don't need to be normal, but these are.

In this problem you do not know the population standard deviation.

The mean for the first set ¯xx¯ ₁ = 18.831 with a standard deviation of s₁ = 1.48 There sample size was 12.

The mean for the first set ¯xx¯  2  = 32.062 with a standard deviation of s₂ = 1.035 There sample size was 21.

Find the degrees of freedom.   (round to 2 places)

Find the left side critical value for a two tail test, t-star for alpha = 0.01. Use the truncated version of the degrees of freedom.   (Remember the right tail is just the positive version.)

Find the test statistic t=   Round to 4 places.

Answer 1

A)

Here given that the population are normally distributed

And Given that ,

sample mean M = 60.1

sample standard deviation = σ = 14.5

sample size =n = 36

here we do not know the population standard deviation

To test :

H₀ :  µ = µ0 versus   H_a : µ < µ0

where µ0 specified population mean = 62.3

therefore ,

To test :

H₀ :  µ = 62.3 versus   H_a : µ < 62.3

Test Statistics :

z=-0.910

Test statistic z = -0.910

given significance level of α=0.01

critical value = z_α = z_0.01 = -2.326

Decision : If z < z_{α ,} then we reject H0 at 0.01 significance level

here z_calculate  =-0.910 >  z_α = -2.326 at 0.01 significance level Then we do not reject the null hypothesis.

Conclusion :It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean M  is less than 62.3, at the 0.01 significance level.

As such, the final conclusion is that...

There is not sufficient evidence to warrant rejection of the claim that the population mean is less than 62.3.

B)

In this case we use t test

In this problem you do not know the population standard deviation.

given sample information

For data set 1

sample mean =

sample standard deviation s1 = 1.48

sample size n1 = 12

for data set 2

sample mean =

sample standard deviation s2 =1.035

sample size n2 =21

To Test :

H₀= µ₁ = µ₂ versus   H_a : µ₁ ≠ µ₂

Test statistic :

t = - 27.3784

here n1 = 12 , n2 = 21

then degrees for freedom = 17.27

hence we find the critial value for left tailed test is

t _{0.01, 17.27} = 2.893

Decision : If | t | > t_critical then we reject H0 at significance level α

here | t | = 27.3784 > t_critica = 2.893 , then we reject null hypothesis at 0.01 significence level

It is concluded that the null hypothesis Ho is rejected.

Conclusion : Therefore, there is enough evidence to claim that population mean μ1​ is different than μ2​, at the 0.01 significance level.