randomly selected people were asked how long they

Eng saciid ahmed

Eng saciid ahmed

Answered question

2022-10-02

randomly selected people were asked how long they slept at night. The mean time was 7.1 hours, and the standard deviation was 0.78 hour. Find the 95% confidence interval of the mean time. Interpret your result.
 

Answer & Explanation

Don Sumner

Don Sumner

Skilled2023-05-29Added 184 answers

To find the 95% confidence interval for the mean time people slept at night, we can use the formula:
Confidence Interval=Sample Mean±(Critical Value)×(Standard DeviationSample Size)
Given that the sample mean is 7.1 hours, the standard deviation is 0.78 hour, and we don't know the sample size, we need to determine the critical value for a 95% confidence level. The critical value corresponds to the level of confidence and the shape of the sampling distribution. Since the sample size is unknown, we'll assume a large enough sample size for the Central Limit Theorem to apply, which allows us to use the Z-distribution.
The critical value for a 95% confidence level with a two-tailed test is approximately 1.96.
Substituting the values into the formula, we get:
Confidence Interval=7.1±(1.96)×(0.78Sample Size)
Interpretation:
The 95% confidence interval for the mean time people slept at night is obtained by taking the sample mean of 7.1 hours and adding/subtracting the margin of error, which is calculated as 1.96 times the standard error of the mean. The margin of error represents the range within which we are reasonably confident that the true population mean lies.

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