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Chapter 9: Problem 31

GPAs (Example 11) In finding a confidence interval for a random sample of 30 students GPAs, one interval was \((2.60,3.20)\) and the other was \((2.65,3.15)\). a. One of them is a \(95 \%\) interval and one is a \(90 \%\) interval. Which is which, and how do you know? b. If we used a larger sample size \((n=120\) instead of \(n=30\) ). would the \(95 \%\) interval be wider or narrower than the one reported here?

Short Answer

Expert verified
The confidence interval (2.60,3.20) is 95% interval and the confidence interval (2.65,3.15) is 90% interval. If the sample size increases from 30 to 120, the 95% confidence interval would be narrower than the ones provided.

Step by step solution

01

Identify confidence intervals

The given two intervals are (2.60,3.20) and (2.65,3.15), confidence interval with greater range should be associated with higher degree of confidence as greater uncertainty is involved. Therefore, (2.60,3.20) is likely to be the 95% interval and (2.65,3.15) is the 90% interval.
02

Effect of sample size on confidence interval

With increase in sample size, the confidence interval becomes narrower assuming the confidence level remains the same. This is due to the inverse relationship between sample size and standard error (precisely, standard deviation divided by square root of sample size, which decreases with increase in sample size). Therefore, if we used a larger sample size (n=120 instead of n=30), the 95% interval would be narrower than the one reported here.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Size Effect
When we collect data to make inferences about a population, the sample size plays a critical role in determining the accuracy of our estimates. A larger sample size generally means a more precise estimate of the population parameters.

Here's why:
  • As the sample size increases, the estimates we make tend to become more reliable because we capture more information about the population.
  • With a larger sample, we reduce the variability of our estimate, leading to a narrower confidence interval.
In the provided example, increasing the sample size from 30 students to 120 would result in a narrower confidence interval at the same confidence level. This is because the standard error decreases as the sample size increases. Therefore, a larger sample allows for a more precise range around the population mean.
Standard Error
The concept of Standard Error is key to understanding how confidence intervals shrink or expand. It represents the average distance that the sample mean of your data will be from the actual population mean.

The standard error can be calculated using the formula:\[SE = \frac{\sigma}{\sqrt{n}}\]where \(\sigma\) is the population standard deviation, and \(n\) is the sample size. As you can see, the standard error inversely relates to the sample size:
  • Larger samples produce a smaller standard error, which in turn leads to a narrower confidence interval.
  • This is because larger samples better approximate the population, reducing the influence of random sampling variability.
The smaller the standard error, the more our sample mean is tightly clustered around the true population mean, making our confidence intervals more precise.
95% vs. 90% Confidence Levels
Confidence level indicates how sure we are that the true population parameter lies within the confidence interval we calculated. Higher confidence levels mean wider intervals, capturing more potential variability in the sample data.

In our example:
  • The interval (2.60, 3.20) is wider, hence it is the 95% confidence interval. This suggests a higher degree of certainty but also more room to account for potential data variability.
  • The interval (2.65, 3.15) is narrower, likely corresponding to the 90% confidence interval, which implies less certainty but a more precise estimate.
Choosing between these intervals depends on the researcher鈥檚 willingness to trade precision for certainty. A 95% interval suggests we are very confident the true mean is within that range, while a 90% interval offers a slightly less certain, yet more precise, range.

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Most popular questions from this chapter

Potatoes Use the data from Exercise \(9.35\). a. If you use the four-step procedure with a two-sided alternative hypothesis, should you be able to reject the hypothesis that the population mean is 20 pounds using a significance level of \(0.05 ?\) Why or why not? The confidence interval is reported here: I am \(95 \%\) confident that the population mean is between \(20.4\) and \(21.7\) pounds. b. Now test the hypothesis that the population mean is not 20 pounds using the four-step procedure. Use a significance level of \(0.05\). c. Choose one of the following conclusions: i. We cannot reject a population mean of 20 pounds. ii. We can reject a population mean of 20 pounds. iii. The population mean is \(21.05\) pounds.

Exam Scores The distribution of the scores on a certain exam is \(N(70,10)\), which means that the exam scores are Normally distributed with a mean of 70 and standard deviation of \(10 .\) a, Sketch the curve and label, on the \(x\) -axis, the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score will be between 50 and 90 . Shade the region under the Normal curve whose area corresponds to this probability.

Oranges A statistics instructor randomly selected four bags of oranges, each bag labeled 10 pounds, and weighed the bags. They weighed \(10.2,10.5,10.3\), and \(10.3\) pounds. Assume that the distribution of weights is Normal. Find a \(95 \%\) confidence interval for the mean weight of all bags of oranges. Use technology for your calculations. a. Decide whether each of the following three statements is a correctly worded interpretation of the confidence interval, and fill in the blanks for the correct option(s). i. I am \(95 \%\) confident that the population mean is between ii. There is a \(95 \%\) chance that all intervals will be between iii. I am \(95 \%\) confident that the sample mean is between b. Does the interval capture 10 pounds? Is there enough evidence to reject the null hypothesis that the population mean weight is 10 pounds? Explain your answer.

Exam Scores The distribution of the scores on a certain exam is \(N(70,10)\), which means that the exam scores are Normally distributed with a mean of 70 and standard deviation of \(10 .\) a. Sketch the curve and label, on the \(x\) -axis, the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score will be bigger than 80\. Shade the region under the Normal curve whose area corresponds to this probability.

Babies Weights (Example 2) Some sources report that the weights of full-term newborn babies have a mean of 7 pounds and a standard deviation of \(0.6\) pound and are Normally distributed. a. What is the probability that one newborn baby will have a weight within \(0.6\) pound of the mean-that is, between \(6.4\) and \(7.6\) pounds, or within one standard deviation of the mean? b. What is the probability the average of four babies' weights will be within \(0.6\) pound of the mean; will be between \(6.4\) and \(7.6\) pounds? c. Explain the difference between a and \(\mathrm{b}\).
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