# Statistics Problems

6.10 Margin of error and the confidence interval.

A study based on a sample of size 25 reported a mean of 93 with a margin of error of 11 for 95% confidence.

(a) Give the 95% confidence interval.

(b) If you wanted 99% confidence for the same study, would your margin of error be greater than, equal to, or less than 11? Explain your answer.

6.20 Apartment rental rates.

You want to rent an unfurnished one- bedroom apartment in Boston next year. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $ 1400. Assume that the standard deviation is $ 220. Find a 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community.

6.26 Fuel efficiency.

Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon ( mpg). For one vehicle equipped in this way, the mpg were recorded each time the gas tank was filled, and the computer was then reset. 10 Here are the mpg values for a random sample of 20 of these records:

41.5 50.7 36.6 37.3 34.2 45.0 48.0 43.2 47.7 42.2 43.2 44.6 48.4 46.4 46.8 39.2 37.3 43.5 44.3 43.3 Suppose that the standard deviation is known to be s = 3.5 mpg.

(a) What is sx, the standard deviation of x?

(b) Give a 95% confidence interval for µ, themean mpg for this vehicle.

6.32 Accuracy of a laboratory scale.

To assess the accuracy of a laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly. The scale readings are Normally distributed with unknown mean ( this mean is 10 grams if the scale has no bias). The standard deviation of the scale readings is known to be 0.0002 gram.

(a) The weight is measured five times. The mean result is 10.0023 grams. Give a 98% confidence interval for the mean of repeated measurements of the weight.

(b) How many measurements must be averaged to get a margin of error of ± 0.0001 with 98% confidence?

6.50 What's wrong?

Here are several situations where there is an incorrect application of the ideas presented in this section. Write a short paragraph explaining what is wrong in each situation and why it is wrong.

(a) A random sample of size 20 is taken from a population that is assumed to have a standard deviation of 12. The standard deviation of the sample mean is 12/ 20.

(b) A researcher tests the following null hypothesis: H0: x = 10.50

(c) A study with x = 48 reports statistical significance for Ha: µ > 54.

(d) A researcher tests the hypothesis H0: µ = 50 and concludes that the population mean is equal to 50.

6.56 Computing the P- value.

A test of the null hypothesis H0: µ = µ0 gives test statistic z = 1.34.

(a) What is the P- value if the alternative is Ha: µ > µ0?

(b) What is the P- value if the alternative is Ha: µ < µ0?

(c) What is the P- value if the alternative is Ha: µ = µ0?

6.58 A two- sided test and the confidence interval.

The P- value for a two- sided test of the null hypothesis H0: µ = 30 is 0.04.

(a) Does the 95% confidence interval include the value 30? Why?

(b) Does the 90% confidence interval include the value 30? Why?

6.70 Calcium level in pregnant women in rural Guatemala.

The level of calcium in the blood in healthy young adults varies with mean about 9.5 milligrams per deciliter and standard deviation about s = 0.4. A clinic in rural Guatemala measures the blood calcium level of 160 healthy pregnant women at their first visit for prenatal care. The mean is x = 9.57. Is this an indication that the mean calcium level in the population from which these women come differs from 9.5?

(a) State H0 and Ha.

(b) Carry out the test and give the P- value, assuming that s = 0.4 in this population. Report your conclusion.

(c) Give a 95% confidence interval for the mean calcium level µ in this population. We are confident that µ lies quite close to 9.5. This illustrates the fact that a test based on a large sample ( n = 160 here) will often declare even a small deviation from H0 to be statistically significant.

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6.10

Margin of error and the confidence interval: A study based on a sample of size 25 reported a mean of 93 with a margin of error of 11 for 95% confidence.

(a) Give the 95% confidence interval.

(b) If you wanted 99% confidence for the same study, would your margin of error be greater than, equal to, or less than 11? Explain your answer.

Solution:

The 95% confidence interval is given as follows:

(93-11,93+11)

=(82,104)

If we wanted 99% confidence for the same study, the margin of error would have been greater than 11.

Margin of Error,E =( z_(α/2)×σ)/√n

As the level of confidence increases, the value of the z-critical increases.

z_(α/2) for 95% confidence level=1.96

z_(α/2) for 99% confidence level=2.58

Hence with increase in confidence level, the margin of error increases. 99% confidence interval for the mean implies that we are 99% certain that μ lies within the given interval, and this increase in certainty is due to an increase in the interval width or the margin of error.

6.20

Apartment rental rates: You want to rent an unfurnished one- bedroom apartment in Boston next year. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $ 1400. Assume that the standard deviation is $ 220. Find a 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community.

Solution:

Given:

Sample Size,n=10

Mean,x ̅=1400

Standard Deviation,s=220

The confidence interval for significance level, alpha = 0.05, or in other words, the 95% confidence interval for the mean score is given as:

(x ̅-(z_(α/2)×σ)/√n,x ̅+(z_(α/2)×σ)/√n)

where,Sample Size,n=10

Mean,x ̅=1400

Standard Deviation,s=220

z_(α/2) for 95% confidence=1.96

The 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community is given ...

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This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Statistics and provides students with a clear perspective of the underlying concepts.