Question

The claim is that the IQ scores of statistics professors are normally​ distributed, with a mean less than 133. A sample of 11 professors had a mean IQ score of 130 with a standard deviation of 8. Find the value of the test statistic.

Answer 1

SOLUTION:

From given data,

The claim is that the IQ scores of statistics professors are normally​ distributed, with a mean less than 133. A sample of 11 professors had a mean IQ score of 130 with a standard deviation of 8. Find the value of the test statistic.

Test : < 133 with t distribution

Hypothesis test:

H₀ : = 133 (Null Hypothesis , H_a : >133 is also correct )

H_a : < 133 ( Alternative hypothesis, also called H₁ )

This is lower tailed test:

= 130

standard deviation = s = 8

sample = n = 11

Let us assume Significance Level,

= 0.05

Degree of freedom (df) = n-1 = 11-1 = 10

Test statics ,

t_observed = ( - ) / ( s / )

= (130  - 133) / ( 8 / )

= -1.243

Using t-table , we can observe that the p - value is between 0.005 and 0.01

Using excel we find p -value = P(t <  -1.243 ) = t.dist(-1.243 , 10 ,1) = 0.121111

P-value = 0.121111

t_critical = - t_,df  = - t_0.05,10 = - 1.812461 ( from t-table , one - tailed , df = 11 )

Critical value =  -1.812  

Rejection criteria : Rejection region is direction of Alternative hypothesis , since

H_a : < 133 , rejection rule is:

Reject H₀ if test statistic , t₀ < -1.812  

Decision :

Since   -1.25 <   - 1.812 , we reject the null hypothesis ,

Hence at 5% significance level , we have sufficient evidence to conclude that   is less than 133