EXERCISE - A
2) A mechanist is making engine parts with axle diameter of 0.7 inch. A random sample of 10 parts shows mean diameter 0.742 inch with a standard division of 0.04 inch. On the basis of this sample, would you say that the work is inferior ? (Given t₉(0.05)= 2.262). 3.15, the null hypothesis is rejected at 5% level of significance or the alternative hypothesis is accepted at 5% level of significance. Hence , sample mean differs significantly from population means e.g, the work is failure
3) A soap manufacturing company was distributing a particular brand of a soap through a large number of retail shops. Before a heavy advertisement campaign, the mean sales per week per shop was 140 dozens . After the campaign a sample of 26 shops was taken and mean sales was found to be 147 dozens with standard deviation 16. Can you considere the advertisement effective ?( given t₂₅(0.05)= 2.06). 2.187, we reject the null hypothesis and accept the alternative hypothesis. Hence, we conclude that advertisement is effective for sales .
4) A random sample of size 16 has 53 as mean. The sum of the squares of the divisions taken from mean is 150. Can this sample be regarded as taken from the population having 56 as mean ? given t₁₅(0.01)= 2.95). -3.794, we reject the null hypothesis. Consequently, the alternative hypothesis accepted at 0.01 level of significance. Hence , the sample is not taken from the population having 56 as mean .
5) A random sample of 17 values from a normal population has a mean of 105cm and the sum of the squares of the deviations from this mean is 1225cm². Is the assumption of a mean of 110cm for the normal population reasonable ? Test under 5% and 1% levels of significance. Also, obtain the 95% and 99% confidence limits. (Given t₁₆(0.05)= 2.11) and t₁₆(0.01)= 2.921). Yes at 1% level
6) Ten students are selected at random from a college and their heights are found to be 100, 104, 108, 110,118, 120, 122, 124, 126 and 128cms. In the light of these data, discuss the suggestion that the mean height of the Student of the college is 110cms (given t₉(0.05)= 2.262). Yes
7) A random sample of 10 boys had the following I. Q's: 70, 120, 110, 88, 83, 95, 98, 107, 100. Do these data support the assumption of a population mean IQ of 100 ? Find a reasonable range in which most of them IQ. values of sample of 10 boys lie. (Given t₉(0.05)= 2.262)
Miscellaneous -1
1) Ten cartons are taken at random from an automatic filling machine. The mean net weight of the cartons is 11.8 kg and the standard deviation 0.15 kg. Does the sample mean differ significantly from the intended weight of 12 kg? (Given t₉(0.05)= 2.262). Yes
2) A machine is designed to produce insulating washers for electrical devices of average thickness of 0.025cm. A random sample of 10 washers was found to have an average thickness of 0.024 cm with a standard deviation of 0.002cm. test the significance of the deviation (Given t₉(0.05)= 2.262). Not significant
3) A random sample of size 25 from a normal population has the mean 47.5 and standard deviation 8.4. Does this information refute the claim that the mean of the population is 42%. (Given t₂₄(0.05)= 2.06). Yes
4) A process of marketing certain bearings is under control if the diameter of the bearings have the mean 0.5cm. What can we say about this process if a sample of 10 of these bearings has a mean diameter of 0.506 cm and standard deviation of 0.004 cm ? (Given t₉(0.05)= 2.262). Process is not under control
5) A machine is supposed to produce washers of mean thickness 0.12cm. A sample of 10 washers was found to have a mean thickness of 0.128 and standard deviation 0.008. test whether the machine is working in proper order at 5% level of significance. (Given t₉(0.05)= 2.262). No
6) A random sample of 16 values from a normal population showed a mean of 41.5 and sum of squares of deviations from mean equal to 135. Can it be assumed that the mean of the population is 43.5 ? (Given t₁₅(0.01)= 2.95). Yes
7) A sample of size 9 from a normal population mean= 15.8 and s²= 10.3. Find 99% confidence interval for the population mean. (Given t₈(0.01)= 3.335). (11.99,19.61)
8) A random sample of size 16 has 53 as mean. The sum of the squares of deviations taken from mean is 150. Find 95% and 99% confidence interval for population mean. (Givent₁₅(0.01)= 2.95 and t₁₅(0.05)= 2.13). (51.32,54.68),(50.67,55.33)
9) A random sample of 16 values from a normal population showed a mean of 41.5 inches and the sum of squares of deviations from this mean equal to 135 square inches. Show that the assumption of a mean of 43.5 inches for a population is not reasonable. Obtained 95% and 99% confidence intervals for the same. (Given t₁₅(0.05)=2.131) and t₁₅(0.01)= 2.947). (39.902,43.098),(39.29,43.71)
10) The annual rainfall at a certain place is normally distributed with mean 45cm. The rainfall during the last 5 years are 48cm, 42 cm, 40cm, 44cm and 43cm. Can we conclude that the average rainfall during the last 5 years is less than the normal rainfall ? (Given t₄0.05)= 2.132). No
11) The height of 8 males participating in an athletic championship are found to be 175 cm, 168cm, 170cm, 167cm, 160cm, 173cm and 168 cm. Can we conclude that the average height is greater than 165cm ?(given t₇(0.05)= 1.895). Yes
12) The mean weakly sells of chocolate bar in general stores was 146.3 bars per store. After an advertising the mean weekly sales in 22 stores for typical week increased to 153.7 bars and showed a standard deviation of 17.2. Was the advertising campaign successful ? (Given t₂₁(0.05)= 2.08). Yes
13) The foreman of ABC mining company has estimated the average quantity of iron ore extracted to be 36.8 tonnes per shift and the sample standard deviation to be 2.8 tonnes per shift, based upon a random selection of 4 shifts. Consider a 90% confidence interval around this estimate. (Given t₃(0.1)= 2.353). (34.5, 39.10)
14) A random sample of size 20 from a normal population gives a sample mean of 42 and standard division of 6. Test the hypothesis that the population mean is 44. (give t₁₉(0.05)= 2.09). True
15) A random sample of 10 boys had the following I. Q: 70, 120, 110, 101, 88, 83, 95, 93, 107, 100. Do thesr data support the assumption of a population mean IQs of 100? (Given t₉(0.05)=2.262). also, find the 95% confidence interval for the population mean. Yes (87.494, 107.906)
16) The manufacturer of a certain make of electric bulbs claims that his bulbs have a mean life of 25 months with a standard deviation of 5 months. A random sample of 12 such bulbs gave the following values:
life in months: 24 26 32 28 20 18 23 27 29 34 20 28
Can you regard the producer's claim to be valid at 1% level of significance ? Yes
EXERCISE - B
1) For the following data examination if the means of two samples differ significantly:
size means standard deviation
sample I: 6 40 8
sample II: 5 50 10
(Given t₉(0.05)=2.262). is not differ significantly
2) Two batches of the same product are tested for their mean life. Assumption that the lives of the product follows a normal distribution with an unknown variance, test the hypothesis that the mean life is the same for both the branches, given the following information :
Batch sam-size mean life(hrs) sd
I 10 750 12
II 8 820 14
(Use t₁₆+0.05)=2.2120). Yes
3) Samples of two types of electric light bulbs were tested for length of life and following data were obtained:
Type-I. Type -II
S size: 8 7
S. means : 1234 1036
S. SD 36 40
Is the difference in the mass sufficient to warrant that type I is superior to type II regarding length of life ? (Given t₁₃(0.05)= 2.216). No
4) Two different types of drugs A and B were tried on certain patients for increasing weight,5 persons were given A 7 persons were given drug B. The increase the weights in pounds is given below:
Drug A: 8 12 13 9 3
Drug B: 10 8 12 15 6 8 11
Do the two drugs differ significantly with regard to their effect in increasing the weight (given t₁₀(0.05)= 2.23). No
5) The height in inches of 6 randomly chosen sailors and 10 randomly chosen soldiers are given as under :
Sailors Soldiers
63 61
65 62
68 65
69 66
71 69
72 69
70
71
72
73
Does these figures show that the soldiers are on an average shorter than sailors? (Given t₁₄(0.05)= 2.15). Yes
6) The mean life of a sample of 10 bulbs was found to be 1456 hours with standard deviations of 423 hours. A second sample of 17 bulbs chosen from a different batch showed a mean in life of 1280 hours with standard deviation of 398 hours. Is there a significant difference between the means of the two batches ? No
7) Strength tests carried out on samples of 2 years spun to the same count gave the following results:
Size of sample samplemean Sam-size sample vari.
Yarn A 4 50 42
Yarn B 9 42 56
The strengths are expressed in kg. Is the difference in mean strengths significant of real difference in the mean strengths of the sources from which samples are drawn? Not significant
8) Samples of two types of electric bulbs were tested for length of life and the following data were obtained:
Type I Type II
No. Of bulbs in the sample 8 7
Mean(in hours) 1134 1024
S. D(in hours): 35 40
Is the difference in the sample means significant? (Given t₁₃(0.05)= 2.16). Significant
9) I. Q test on two groups of boys and girls gave the following results:
Girls: mean=78 n= 50 s.d= 10
Boys : mean=73 n= 100 s.d= 15
Is there a significant difference in the mean scores of boys and girls? (Given t₂₃(0.05)= 2.07)
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