III. Testing the sample mean:
Let us consider random sample x1, x2, ............ xi, ... xn of size n which is assumed to be drawn from a normal population.
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We know that the sample mean.
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Let us consider.
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We frame the null hypothesis:
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OR
H_0: There is no significant difference between sample mean and population mean.
~ H_1: There is significant difference between sample mean and population mean.
To test H_0 we use the Z statistic.
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If Sigma is unknown, then we use the sample standard deviation 's'.
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Confidence Interval of the population mean:
To find the confidence interval we use the condition for accepting the null hypothesis.
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Let us consider.
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Combine (1) and (2), we get.
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Problems:
1. Random sample of 100 students selected from a university has the mean IQ of 110 with the standard deviation 5 can use conclude that mean IQ of the students is 120.
Given that.
The sample size, sample mean, sample standard deviation, population mean are.
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OR
H_0: The claim is valid ~ H_1: The claim is invalid.
To test the H_0 we use the Z - Statistics.
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2. A company specified that average life of the electric bulbs is 1600 hours. The random sample of 100 bulbs produced by the company has an average lifetime of 1570 hours with standard deviation 120 hours. Test the manufactures claim at 5% level of significance.
Given that
The sample size, sample mean, population mean, sample standard deviation as.
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OR
H_0 : The claim is valid. ~ H_1 : The claim is invalid.
To test H_0 we use the Z Statistics.
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IV. Testing of difference between two sample means:
Let us consider two independent samples of sizes n1 and n2 denoted by x1, x2,.........xi,.....xn1 and y1, y2,.......yj.....yn2.
The first sample mean as
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The second sample mean as
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Let us assume that.
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The sampling distribution of difference between means is.
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We frame the null hypothesis.
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OR
H_0: There is no significant difference between the population means.
~ H_1: There is significant difference between the population means.
To test H_0 we use the Z Statistic.
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If the population values are unknown then we assume that,
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Where,
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We compare |Z cal| with the table value Z alpha. If | Zcal| < Z alpha then we accept H_0 and If |Z Cal| > Z alpha then we reject H_0.
Problems:
1. Is there any significant difference between the average life of two brands of electric bulbs from the following data.
Given that,
The size of the sample 1,
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The size of the sample 2,
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The mean of sample 1,
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The mean of sample 2,
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The standard deviation of sample 1,
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The standard deviation of sample 2,
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OR
H_0: There is no significant difference between the population means ~ H_1: There is significant difference between the population mean.
To test the H_0 we use the Z - Statistic.
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Where,
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Since |Z cal| > 1.98 then we reject H_0 at 5% level of significance. There is significant difference between the population mean or the lives of two brands of electric bulbs are not equal.
2. The average hourly wage of a sample of 150 workers in a company was Rs 256 with standard deviation of Rs 108. The average hourly wage of a sample of 200 workers in another company was 287 with a standard deviation of Rs 128 is there any significant difference between the average hourly wages of the employees in the two companies.
Given that.
The sample size one as,
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The sample size two as,
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The mean of the first sample,
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The mean of the second sample,
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The standard deviation of the first sample,
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The standard deviation of the second sample,
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We frame the null hypothesis:
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OR
H_0: There is no significant difference between the two population means. ~ H_1: There is significant difference between the two population means.
To test H_0 we use the Z statistic.
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Where,
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Since |Z cal| > 1.96 we reject H_0 at 5% level of significance. There is significant difference between the average hourly wages of the employees in the two companies.
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