Question 1

In testing  vs.  the following sample data were recorded: 5.0, 6.1, and 11.1. Then, the value of the test statistic is:

Accepted Answer

The test statistic for a two-sample t-test with unequal variances is calculated as:

t = (x̄1 - x̄2) / s_pooled * sqrt(1/n1 + 1/n2)

where x̄1 and x̄2 are the sample means, n1 and n2 are the sample sizes, and s_pooled is the pooled standard deviation calculated as:

s_pooled = sqrt(((n1 - 1)s1^2 + (n2 - 1)s2^2) / (n1 + n2 - 2))

where s1 and s2 are the sample standard deviations.

Using the given data, we have:

x̄1 = 7.4
x̄2 = 7.4
n1 = n2 = 3
s1 = 3.011
s2 = 2.113

Plugging these values into the formula for s_pooled, we get:

s_pooled = sqrt(((3 - 1)(3.011)^2 + (3 - 1)(2.113)^2) / (3 + 3 - 2)) = 2.616

Plugging in all the values, we get:

t = (7.4 - 7.4) / 2.616 * sqrt(1/3 + 1/3) = 0 / 1.508 = 0

Therefore, the test statistic is 0, which corresponds to the null hypothesis that there is no difference between the two groups.

The closest answer choice to 0 is B) 5.285, which is incorrect. Therefore, the answer is E) none of these.

Question 2

If a hypothesis test is to be conducted regarding a population variance, then the critical value of the test will be obtained using the:

Accepted Answer

When conducting a hypothesis test regarding a population variance, we use the chi-square distribution with n-1 degrees of freedom to obtain the critical value of the test.

Question 3

A pasta company would like to know the variability in the shelf life of their product. A random sample of 20 packages of pasta yielded a standard deviation of 5 days. Estimate the population variance using a 90% confidence interval. Assume the distribution is normal.
______________ Enter (n1, n2)

Accepted Answer

The answer of A pasta company would like to know...

Question 4

A random sample of 20 observations is selected from a normally distributed population. The sample variance is 12. In the 95% confidence interval for the population variance, the upper limit will be:

Accepted Answer

We use the chi-square distribution with n-1 degrees of freedom to calculate the confidence interval.

The interval is given by:

( (n-1)*s^2 ) / chi-squared(alpha/2, n-1) <= sigma^2 <= ( (n-1)*s^2 ) / chi-squared(1-alpha/2, n-1)

where s^2 is the sample variance, n is the sample size, alpha is the significance level, and chi-squared is the chi-square distribution function.

Substituting the values gives:

(19*12) / chi-squared(0.025, 19) <= sigma^2 <= (19*12) / chi-squared(0.975, 19)

Using a chi-square table or calculator, we find that:

chi-squared(0.025, 19) = 32.8523
and 
chi-squared(0.975, 19) = 8.9065

Substituting and simplifying gives:

140.16 / 32.8523 <= sigma^2 <= 140.16 / 8.9065
4.2651 <= sigma^2 <= 15.7328

The upper limit of the interval is 15.7328, which corresponds to choice letter D, 25.599.

Question 5

A random sample of size 20 taken from a normally distributed population resulted in a sample variance of 32. The lower limit of a 90% confidence interval for the population variance would be:

Accepted Answer

The lower limit of the 90% confidence interval for the variance can be found using the chi-square distribution with degrees of freedom n-1, where n is the sample size.

The chi-square value for a 90% confidence interval and 19 degrees of freedom (20-1) is 29.707.

The lower limit of the confidence interval is then (n-1)*s^2/chi-square value = (19*32)/29.707 = 20.170.

Therefore, the answer is C, 20.170.

Question 6

If you are testing  vs.  using a sample of 15 observation and a significance level equal to .05, then the critical value of the test is:

Accepted Answer

The answer of If you are testing  vs. ...

Question 7

A random sample of 25 observations is selected from a normally distributed population. The sample variance is 10. In the 95% confidence interval for the population variance, the upper limit will be:

Accepted Answer

The upper limit of the confidence interval for the population variance is calculated using the chi-square distribution. With 24 degrees of freedom (n-1) and a 95% confidence level, the chi-square value for the upper limit is approximately 12.401. The upper limit is calculated as (24 * 10) / 12.401 = 19.353.

Question 8

In a hypothesis test for the population variance, the hypotheses are  vs.  . If the sample size is 15 and the test is being carried out at the 10% level of significance, the rejection region will be:

Accepted Answer

The answer of In a hypothesis test for the population...

Question 9

The sampling distribution used to make inferences about a single population variance is:

Accepted Answer

The sampling distribution used to make inferences about a single population variance is a chi-square distribution.

Question 10

The variability in a scientist's measuring equipment was observed for a random sample of 26 test runs from a normal distribution. The sample yielded a variance of 33. Estimate the population variance using a 95% confidence interval.
______________ Enter (n1, n2)

Accepted Answer

The answer of The variability in a scientist's measuring equipment...

Question 11

The statistic  is chi-squared distributed with n-1 degrees of freedom only if:

Accepted Answer

The chi-squared distribution assumes that the population is normally distributed with variance equal to the degrees of freedom. Therefore, choice A is the correct option. Choices B and C are incorrect because they do not mention anything about the population being normally distributed with a specific variance. Choice D is incorrect because it includes choice C, which is not correct. Choice E is also incorrect because the chi-squared distribution has specific assumptions about the population distribution.

Question 12

A car salesperson tries to convince customers buying a new Honda Accord that the gas mileage varies by only 2 miles per gallon when driving in town and on the highway. The customer found, for 14 randomly chosen tanks of gas, the gas mileage to vary by 2.8 miles per gallon. Do the customer's results contradict the salesperson's claim? Test the relevant hypotheses at the 0.05 level of significance. Assume the sample comes from a normal population. Test Statistic = ______________ Reject Region: Reject H₀ if > ______________ Conclusion: ______________ Interpretation: ________________________________________________________

Accepted Answer

The answer of A car salesperson tries to convince customers...

Question 13

A city bus driver claims that the route from the Canyon Crest Town Center to the University of California, Riverside never varies more than 1.3 minutes. A random sample of 18 trips yielded a variance of 1.26 minutes². It is of interest to determine if the sample data present sufficient evidence to reject the driver's claim. Calculate the value of the test statistic. What is the appropriate conclusion at the 0.05 level of significance? Test Statistic = ______________ Reject Region: Reject H₀ if > ______________ Conclusion: ______________ One ______________ conclude that the route varies by more than 1.3 minutes.

Accepted Answer

The answer of A city bus driver claims that the...

Question 14

A random sample of size 25 taken from a normally distributed population resulted in a sample standard deviation of a 0.93054. The lower and upper limits of a 99% confidence interval for the population variance would be:

Accepted Answer

The answer of A random sample of size 25 taken...

Question 15

The shape of the chi-squared distribution depends on:

Accepted Answer

The shape of the chi-squared distribution depends only on the number of its degrees of freedom. It is a theoretical distribution that models the sum of squared deviations from the mean, and its shape changes depending on the degrees of freedom. The population variance and mean do not directly affect the shape of this distribution.

Question 16

The chi-squared distribution is:

Accepted Answer

The chi-squared distribution is positively skewed and ranges between 0 and infinity.

Question 17

Like that of the Student t-distribution, the shape of the chi-squared distribution depends on:

Accepted Answer

The shape of the chi-squared distribution depends on the number of its degrees of freedom. The distribution is skewed to the right and becomes more symmetrical as the degrees of freedom increase. The population size, population standard deviation, and whether the population is unimodal or bimodal are not factors that affect the shape of the chi-squared distribution.

Question 18

Which of the following statements is true with regard to a hypothesis test for a population variance  ?

Accepted Answer

The answer of Which of the following statements is true...

Question 19

The sampling distribution of the quantity  is:

Accepted Answer

The answer of The sampling distribution of the quantity ...

Question 20

In testing  vs.  the following sample data were recorded: 11.5, 6.5, and 5.4. Then, the p-value of the test is:

Accepted Answer

The answer of In testing  vs.  the following...

Quiz 10: Inference From Small Samples

In testing vs. the following sample data were recorded: 5.0, 6.1, and 11.1. Then, the value of the test statistic is:

If a hypothesis test is to be conducted regarding a population variance, then the critical value of the test will be obtained using the:

A pasta company would like to know the variability in the shelf life of their product. A random sample of 20 packages of pasta yielded a standard deviation of 5 days. Estimate the population variance using a 90% confidence interval. Assume the distribution is normal. ______________ Enter (n1, n2)

A random sample of 20 observations is selected from a normally distributed population. The sample variance is 12. In the 95% confidence interval for the population variance, the upper limit will be:

A random sample of size 20 taken from a normally distributed population resulted in a sample variance of 32. The lower limit of a 90% confidence interval for the population variance would be:

If you are testing vs. using a sample of 15 observation and a significance level equal to .05, then the critical value of the test is:

A random sample of 25 observations is selected from a normally distributed population. The sample variance is 10. In the 95% confidence interval for the population variance, the upper limit will be:

In a hypothesis test for the population variance, the hypotheses are vs. . If the sample size is 15 and the test is being carried out at the 10% level of significance, the rejection region will be:

The sampling distribution used to make inferences about a single population variance is:

The variability in a scientist's measuring equipment was observed for a random sample of 26 test runs from a normal distribution. The sample yielded a variance of 33. Estimate the population variance using a 95% confidence interval. ______________ Enter (n1, n2)

The statistic is chi-squared distributed with n-1 degrees of freedom only if:

A random sample of size 25 taken from a normally distributed population resulted in a sample standard deviation of a 0.93054. The lower and upper limits of a 99% confidence interval for the population variance would be:

The shape of the chi-squared distribution depends on:

The chi-squared distribution is:

Like that of the Student t-distribution, the shape of the chi-squared distribution depends on:

Which of the following statements is true with regard to a hypothesis test for a population variance ?

The sampling distribution of the quantity is:

In testing vs. the following sample data were recorded: 11.5, 6.5, and 5.4. Then, the p-value of the test is: