The brightness of television picture tube can be evaluated b

The brightness of television picture tube can be evaluated by measuring the amount of current required to achieve a particular brightness level. An engineer things that one has to use 300 microamps of current to achieve the required brightness level.

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The brightness of television picture tube can be evaluated by measuring the amount of current required to achieve a particular brightness level. An engineer things that one has to use 300 microamps of current to achieve the required brightness level. A sample of size n 20 has been taken to verify the engineer\'s hypotheses. (a) Formulate the null and the alternative hypotheses. Use two-sided test alternative. (b) For the sample of size n 20 we obtain 319.2 and s = 18.6. Test the hypotheses from part (a) with (c) Use data from part (b), to construct a 95% confidence interval for the mean current required.

Solution

a)
Set Up Hypothesis
Null, one has to use 300 microamps of current to achieve the required brightness level. H0: U=300
Alternate, H1: U!=300
b)
Test Statistic
Population Mean(U)=300
Sample X(Mean)=319.2
Standard Deviation(S.D)=18.6
Number (n)=20
we use Test Statistic (t) = x-U/(s.d/Sqrt(n))
to =319.2-300/(18.6/Sqrt(19))
to =4.616
| to | =4.616
Critical Value
The Value of |t | with n-1 = 19 d.f is 2.093
We got |to| =4.616 & | t | =2.093
Make Decision
Hence Value of | to | > | t | and Here we Reject Ho
P-Value :Two Tailed ( double the one tail ) -Ha : ( P != 4.6164 ) = 0.0002
Hence Value of P0.05 > 0.0002,Here we Reject Ho

We does n\'t have evidence to indicate that one has to use 300 microamps of current to achieve the required brightness level.

c)
Confidence Interval
CI = x ± t a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
ta/2 = t-table value
CI = Confidence Interval
Mean(x)=319.2
Standard deviation( sd )=18.6
Sample Size(n)=20
Confidence Interval = [ 319.2 ± t a/2 ( 18.6/ Sqrt ( 20) ) ]
= [ 319.2 - 2.093 * (4.159) , 319.2 + 2.093 * (4.159) ]
= [ 310.495,327.905 ]