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Chapter 8: Q96E (page 452)

A random sample of n = 6 observations from a normal distribution resulted in the data shown in the table. Compute a 95% confidence interval for σ2

Short Answer

Expert verified

The 95% confidence interval for σ2is (4, 2689, 65.9044).

Step by step solution

01

Given information

Given data is as follows,

8 2 3 7 11 6

02

Calculating the Confidence interval

The 95% confidence interval can be calculated using the formula,

n-1s2χα22⩽σ2⩽n-1s2χ1-α22

From the given data, n=6ands=3.31

From the table values, at the 0.05 level of significance and 5 degrees of freedom, the value for 5 is 12.832 and

the value forn-1s2χ0.0052⩽σ2⩽n-1s2χ0.9952=100-162140.169⩽σ2⩽100-16267.3276 is 0.8312.

Substitute the values to get the required confidence interval

6-13.31212.8325⩽σ2⩽6-13.3120.83124.2689⩽σ2⩽65.9044

Therefore, the 95% confidence interval forσ2is (4, 2689, 65.9044).

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Most popular questions from this chapter

Question: Forecasting daily admission of a water park. To determine whether extra personnel are needed for the day, the owners of a water adventure park would like to find a model that would allow them to predict the day’s attendance each morning before opening based on the day of the week and weather conditions. The model is of the form

where,

y = Daily admission

x1 = 1 if weekend

0 otherwise

X2 = 1 if sunny

0 if overcast

X3 = predicted daily high temperature (°F)

These data were recorded for a random sample of 30 days, and a regression model was fitted to the data.

The least squares analysis produced the following results:

with

  1. Interpret the estimated model coefficients.
  2. Is there sufficient evidence to conclude that this model is useful for predicting daily attendance? Use α = .05.
  3. Is there sufficient evidence to conclude that the mean attendance increases on weekends? Use α = .10.
  4. Use the model to predict the attendance on a sunny weekday with a predicted high temperature of 95°F.
  5. Suppose the 90% prediction interval for part d is (645, 1,245). Interpret this interval.

Question: Independent random samples selected from two normal populations produced the sample means and standard deviations shown below.

Sample 1

Sample 2

n1= 17x¯1= 5.4s1= 3.4

role="math" localid="1660287338175" n2= 12x¯2=7.9s2=4.8

a. Conduct the testH0:(μ1-μ2)>10against Ha:(μ1-μ2)≠10. Interpret the results.

b. Estimateμ1-μ2 using a 95% confidence interval

The “winner’s curse” in transaction bidding. In transaction bidding, the “winner’s curse” is the miracle of the winning (or loftiest) shot price being above the anticipated value of the item being auctioned. The Review of Economics and Statistics (Aug. 2001) published a study on whether shot experience impacts the liability of the winner’s curse being. Two groups of a stab in a sealed-shot transaction were compared (1)super-experienced stab and (2) less educated stab. In the super-experienced group, 29 of 189 winning flings were above the item’s anticipated value; 32 of 149 winning flings were above the item’s anticipated value in the less-educated group.

  1. Find an estimate of p1, the true proportion of super educated stab who fell prey to the winner’s curse
  2. Find an estimate of p2, the true proportion of less-educated stab who fell prey to the winner’s curse.
  3. Construct a 90 confidence interval for p1-p2.
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4.132 Suppose xis a random variable best described by a uniform

probability distribution with c= 3 and d= 7.

a. Find f(x)

b. Find the mean and standard deviation of x.

c. FindP(μ-σ≤x≤μ+σ)

To compare the means of two populations, independent random samples of 400 observations are selected from each population, with the following results:

Sample 1

Sample 2

x¯1=5,275σ1=150

x¯2=5,240σ2=200

a. Use a 95%confidence interval to estimate the difference between the population means (μ1−μ2). Interpret the confidence interval.

b. Test the null hypothesis H0:(μ1−μ2)=0versus the alternative hypothesis Ha:(μ1−μ2)≠0 . Give the significance level of the test and interpret the result.

c. Suppose the test in part b was conducted with the alternative hypothesis Ha:(μ1−μ2)≠0 . How would your answer to part b change?

d. Test the null hypothesis H0:(μ1−μ2)=25 versus Ha:(μ1−μ2)≠25. Give the significance level and interpret the result. Compare your answer with the test conducted in part b.

e. What assumptions are necessary to ensure the validity of the inferential procedures applied in parts a–d?
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