Question

Sara is a big hip-hop music fan. Her friend Matt is a big rap music fan. They each have a huge library of songs in their digital music libraries. They each randomly sample 50 songs from their libraries and record the lengths of the songs selected. The average of the selected hip-hop songs was x₁ = 235 seconds with a sample standard deviation of 4 seconds. The average of the selected rap songs was x₂ = 240 seconds with a sample standard deviation of 6 seconds. They would like to know if the average length of hip-hop songs is different than the average length of rap songs.

What are the appropriate null and alternative hypotheses for this test?

A) H₀: μ₁ − μ₂ < 0

H_a: μ₁ − μ₂ > 0

B) H₀: μ₁ − μ₂ = 0

   H_a: μ₁ − μ₂ ≠ 0

    

C) H₀: μ₁ − μ₂ = 0

H_a: μ₁ − μ₂ > 0

D) H₀: μ₁ − μ₂ = 0

H_a: μ₁ − μ₂ < 0

- Calculate the test statistic. (Round your answer to two decimal places.)

- Assume for purposes of this study, the degrees of freedom are 80. At a 5% significance level, what is the critical value for the test? (Round your answer to two decimal places. Just give the value of the critical value, no negative or positive sign)

Answer 1

Solution:

Given ,

$url=https://latex.codecogs.com/gif.latex?%5Cbar%20x_%7B1%7D&t=1596070764&ymreqid=f4764063-0d40-e2b0-1cd9-67000f014b00&sig=VgwUpcgfoQubG6xC9sM_BQ--~D$  = 235 ,   $url=https://latex.codecogs.com/gif.latex?%5Cbar%20x_%7B2%7D&t=1596070764&ymreqid=f4764063-0d40-e2b0-1cd9-67000f014b00&sig=SDRH.2qpckzWXCmv9wU4Nw--~D$ = 240

s₁ = 4 , s₂ = 6

n₁ = 50 , n₂ = 50

1)

H₀: μ₁ − μ₂ = 0

   H_a: μ₁ − μ₂ ≠ 0

2)

The test statistic t is

t =  

1-T2 . VG + 2.

=  

235-240 F2 0 50

= -4.90

Test statistic t = -4.90

3)

d.f. = 80

$\alpha$ = 5% = 0.05

$\alpha$/2 = 0.025

For two tailed test , critical value is

$\pm t_{\alpha/2,df}=\pm t_{0.025,80}=\pm 1.99$

Critical value : 1.99