Answer :
Answer:
33.293 ± 0.01= 33.303 and 33.383,
Step-by-step explanation:
We first need to fit a normal distribution , but neither the mean nor the standard deviation is given . We therefore estimate the sample mean and sample standard deviation s. Using the data we find ∑fx=378 and
∑fx²=1344 so that mean x` = 2.885 or 2.9 and standard deviation s =1.360
x f fx x² fx²
1 27 27 1 27
2 30 60 4 120
3 29 87 9 261
4 21 84 16 336
5 24 120 25 600
∑f=131 ∑fx=378 ∑fx²=1344
Mean = x`= ∑fx/ ∑f= 2.9
Standard Deviation = s= √∑fx²/∑f-(∑fx/∑f)²
s= √1344/131 - (378/131)²
s= √10.26-(2.9)²
s= √10.26- 8.41
s= √1.85= 1.360
Next we need to compute the expected frequencies for all classes and the value of chi square. The necessary calculations for expected frequencies , ei`s ( ei= npi`) where pi` is the estimate of pi together with the value of chi square are shown below.
Categories zi` P(Z<z) pi` Expected Observed
frequency ei Frequency Oi
1 -1.39 0.0823 0.0823 10.78 27
2 -0.66 0.2546 0.1723 22.57 30
3 0.07 0.5279 0.2733 35.80 29
4 0.808 0.7881 0.2602 34.08 21
5 1.54 0.937 0.1489 19.51 24
Next we need to compute the expected frequencies for all classes and the value of chi square. The necessary calculations for expected frequencies , ei`s ( ei= npi`) where pi` is the estimate of pi together with the value of chi square are shown below.
Categories Expected Observed (oi-ei)²/ei
frequency ei Frequency Oi OBSERVED VALUE
1 10.78 27 24.41
2 22.57 30 1.54
3 35.80 29 1.29
4 34.08 21 5.02
5 19.51 24 1.033
Total 131 33.293
There are five categories , we have used the sample mean and sample standard deviation , so the number of degrees of freedom is 5-1-2= 2
The critical region is chi square ≥ chi square (0.001)(2) =9.21
CONCLUSION:
Since the calculated value of chi square =9.21 does not fall in the critical region we are unable to reject our null hypothesis and conclude normal distribution provides a good fit for the given frequency distribution.
Using the Chisquare test statistic relation, the observed value of the Chisquare statistic is 2.09
The observed value of Chisquare can be calculated thus:
- χ² = [tex] \frac{(observed - Expected)^{2}}{Expected} [/tex]
Since the samples are uniformly distributed ;
- Expected value = (27+30+29+21+24)/5 = 131/5 = 26.2
Substituting the Parameters into the equation :
χ² =[tex] \frac{(27 - 26.2)^{2}}{26.2} + \frac{(30 - 26.2)^{2}}{26.2} + \frac{(29 - 26.2)^{2}}{26.2} + \frac{(21 - 26.2)^{2}}{26.2} + \frac{(24 - 26.2)^{2}}{26.2} [/tex]
χ² =[tex] 0.0244 + 0.5511 + 0.2992 + 1.0321 + 0.1847 [/tex]
χ² =[tex] 2.0915 [/tex]
Hence, the Chisquare value is χ² [tex] = 2.09 [/tex]
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