Question

please needed 4-5.2

4-5.2 Repeat Problem 4-5.1 if the sample size is 50 coils, the sample mean is still 115 Ohms, and the sample standard deviation is 10 Ohms.

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4-5.1 The resistance of coils manufactured by a certain company is claimed to have a mean value of resistance of 1 00 Ohms. A sample of 9 coils is taken and it is found that the sample mean is 1 1 5 Ohms and the sample standard deviation is 20 Ohms. a) Is the claim justified if a 95 % confidence level is used? b) Is the claim justified if a 90% confidence level is used?

Answer 1

Before we go on to construct the confidence interval let us know a bit about t-test and the construction of the confidence interval.

4-5.1)

Here given that,

• The resistance of coils manufactured by a certain company is claimed to have a mean value of resistance of 100 Ohms.
• A sample of 9 coils is taken.
• it is found that the sample mean is 115 Ohms and the sample standard deviation is 20 Ohms.

4-5.1)(a) We need to check if the claim is justified if a 95% confidence interval is used. Now the claim to be justified the claimed mean has be within the confidence interval. Let us construct a 95% confidence interval.

Given that,

where c is the confidence coefficient,

Since confidence coefficient c=0.95,

Now,

[From Biometrika Tables Volume I]

Therefore the confidence interval for μ is,

Lower,

Upper,

Hence the 95% confidence interval for μ is given by (99.6267,130.3733).

Now since the claimed mean value 100 Ohms is within the confidence interval the claim is justified.

4-5.1)(b) We need to check if the claim is justified if a 90% confidence interval is used. Now the claim to be justified the claimed mean has be within the confidence interval. Let us construct a 90% confidence interval.

Given that,

where c is the confidence coefficient,

Since confidence coefficient c=0.90,

Now,

[From Biometrika Tables Volume I]

Therefore the confidence interval for μ is,

Lower,

Upper,

Hence the 90% confidence interval for μ is given by (102.6,127.4).

Now since the claimed mean value 100 Ohms is not within the confidence interval the claim is not justified.

4-5.2)

Here given that,

• The resistance of coils manufactured by a certain company is claimed to have a mean value of resistance of 100 Ohms.
• A sample of 50 coils is taken.
• it is found that the sample mean is 115 Ohms and the sample standard deviation is 10 Ohms.

4-5.2)(a) We need to check if the claim is justified if a 95% confidence interval is used. Now the claim to be justified the claimed mean has be within the confidence interval. Let us construct a 95% confidence interval.

Given that,

where c is the confidence coefficient,

Since confidence coefficient c=0.95,

Now,

Now t values with 49 degrees of freedom are not provided in the Biometrika Tables so here we use normal approximation to approximate the values of the t-distribution. Now this approximation is fairly good if n>30.

Now here,

Therefore the confidence interval for μ is,

Lower,

Upper,

Hence the 95% confidence interval for μ is given by (112.1102,117.8898).

Now since the claimed mean value 100 Ohms is not within the confidence interval the claim is not justified.

4-5.2)(b) We need to check if the claim is justified if a 90% confidence interval is used. Now the claim to be justified the claimed mean has be within the confidence interval. Let us construct a 90% confidence interval.

Given that,

where c is the confidence coefficient,

Since confidence coefficient c=0.90,

Now,

Now t values with 49 degrees of freedom are not provided in the Biometrika Tables so here we use normal approximation to approximate the values of the t-distribution. Now this approximation is fairly good if n>30.

Now here,

Therefore the confidence interval for μ is,

Lower,

Upper,

Hence the 90% confidence interval for μ is given by (112.5673,117.4327).

Now since the claimed mean value 100 Ohms is not within the confidence interval the claim is not justified.