Question

In: Statistics and Probability

The claim is that the IQ scores of statistics professors are normally​ distributed, with a mean...

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.

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Expert Solution

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 , Ha : >133 is also correct )

Ha : < 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 ,

tobserved = ( - ) / ( 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

tcritical = - t,df  = - t0.05,10 = - 1.812461 ( from t-table , one - tailed , df = 11 )

Critical value =  -1.812  

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

Ha : < 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


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