Asked by Sushmita Rathour on May 29, 2024
Verified
Two types of flares are tested for their burning times (in minutes) and sample results are given below. Brand X\text { Brand } \mathrm { X } Brand X Brand Y\text { Brand } \mathrm { Y } Brand Y n = 35 n = 40 x‾\overline{x}x = 19.4 x‾\overline{x}x = 15.1 s = 1.4 s = 0.8
Construct a 95% confidence interval for the difference μX\mu \mathrm { X }μX - μY\mu YμY based on the sample data.
A) (3.5,5.1)
B) (3.8,4.8)
C) (3.6,5.0)
D) (-4.7,-3.9)
E) (3.2,5.4)
Burning Times
A term that could refer to the duration or rate at which something burns, though context is needed for a precise definition.
Confidence Interval
An estimated range of values calculated from a given set of sample data that is likely to include the true population parameter.
- Analyze the interpretation of confidence intervals in the context of comparing two population means.
- Recognize and illustrate the implementation of t-tests and significance levels for hypothesis testing purposes.
Verified Answer
AH
abdul hassanJun 03, 2024
Final Answer :
B
Explanation :
To construct the confidence interval for the difference between two means, we use the formula:
(mean1 - mean2) ± t_critical * sqrt((s1^2 / n1) + (s2^2 / n2))
where mean1 and mean2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and t_critical is the critical value from the t-distribution with (n1 + n2 - 2) degrees of freedom for a 95% confidence interval (t_critical = 2.021).
Plugging in the given values, we get:
(19.4 - 15.1) ± 2.021 * sqrt((1.4^2 / 35) + (0.8^2 / 40))
= 4.3 ± 0.494
= (3.806, 4.794)
Therefore, the best choice is B, which matches this interval.
(mean1 - mean2) ± t_critical * sqrt((s1^2 / n1) + (s2^2 / n2))
where mean1 and mean2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and t_critical is the critical value from the t-distribution with (n1 + n2 - 2) degrees of freedom for a 95% confidence interval (t_critical = 2.021).
Plugging in the given values, we get:
(19.4 - 15.1) ± 2.021 * sqrt((1.4^2 / 35) + (0.8^2 / 40))
= 4.3 ± 0.494
= (3.806, 4.794)
Therefore, the best choice is B, which matches this interval.
Learning Objectives
- Analyze the interpretation of confidence intervals in the context of comparing two population means.
- Recognize and illustrate the implementation of t-tests and significance levels for hypothesis testing purposes.
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