Thank you for visiting The high temperatures in degrees Fahrenheit of a random sample of 5 small towns are 99 6 99 7 97 9 98 6 and 97. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Answer: (97.63, 99.77)
Step-by-step explanation:
Given the data:
99.6 99.7 97.9 98.6 97.7
Using calculator, we can obtain the mean and standard deviation of the sample data:
Mean(m) = 98.7
Standard deviation = 0.93
Sample size (n) = 5
Using the relation to find confidence interval :
Mean ± Zcrit * (s/√n)
Zcrit at 99% = 2.576
98.7 ± 2.576 * (0.93 / √5)
Lower limit : 98.7 - (2.576 * 0.4159086) = 97.6286194464 = 97.63 ( 1 decimal place)
Upper limit : 98.7 + (2.576 * 0.4159086) = 99.7713805536 = 99.77 ( 1 decimal place)
(97.6, 99.8)
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Final answer:
A 99% confidence interval for the mean high temperature of these towns is calculated using the sample mean, standard deviation and appropriate t-value. It requires computation of sample mean, standard deviation, standard error and then utilizing these in the formula for the confidence interval.
Explanation:
To create a 99% confidence interval for the mean temperature, we will primarily need two things: the sample mean (μ) and the standard deviation (σ). The sample mean is simply the average of the temperatures you provided, which in this case equals (99.6 + 99.7 + 97.9 + 98.6 + 97.7) / 5 = 98.7 degrees Fahrenheit. The standard deviation can be calculated using the formula for the population standard deviation.
Once we have these, we can use the t distribution to generate our confidence interval, since we know that temperatures are normally distributed and our sample size is relatively small (n<30). For a 99% confidence interval with 4 degrees of freedom (5-1), the t-value roughly equals 3.747, according to the t-distribution table.
The standard error (σ/√n) equals standard deviation divided by the square root of the number of samples. Plugging in the values, we can now compute the confidence interval as follows: μ ± t*(σ/√n).
Learn more about Confidence Interval here:
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