Refer to the Naval Oceanography Portal data on number of...

Refer to the Naval Oceanography Portal data on number of minutes of daylight per day in Sharon, PA, Exercise 7.37 (p. 317$)$. An estimate of the mean number of minutes of daylight per day was obtained using data collected for 12 randomly selected days (one each month) in $2009 .$ a. Determine the number of days in 2009 that need to be sampled in order to estimate the desired mean to within 45 minutes of its true value with $95 \%$ confidence. b. Based on your answer, part a, develop a sampling plan that will likely result in a random sample that is representative of the population. c. Go to the Web site, http://aa.usno.navy.mil/USNO/ astronomical-applications/data-services, and collect the data for Sharon, PA in year 2009 using your sampling plan. d. Use the data, part $\mathbf{c},$ to construct a $95 \%$ confidence interval for the desired mean. Does your interval have the desired width?

Refer to the Naval Oceanography Portal data on number of minutes of daylight per day in Sharon, PA, Exercise 7.37 (p. 317$)$. An estimate of the mean number of minutes of daylight per day was obtained using data collected for 12 randomly selected days (one each month) in $2009 .$
a. Determine the number of days in 2009 that need to be sampled in order to estimate the desired mean to within 45 minutes of its true value with $95 \%$ confidence.
b. Based on your answer, part a, develop a sampling plan that will likely result in a random sample that is representative of the population.
c. Go to the Web site, http://aa.usno.navy.mil/USNO/ astronomical-applications/data-services, and collect the data for Sharon, PA in year 2009 using your sampling plan.
d. Use the data, part $\mathbf{c},$ to construct a $95 \%$ confidence interval for the desired mean. Does your interval have the desired width?

Key Concepts

Sample Size Determination

Sample size determination involves calculating the number of observations needed to achieve a desired level of precision in the estimation of a population parameter, such as ensuring that the margin of error does not exceed a specified amount. This calculation takes into account factors like the population variability, the desired confidence level, and the acceptable margin of error.

Representative Sampling

Representative sampling is the practice of designing a sample in such a way that it accurately reflects the characteristics of the overall population. This is achieved by ensuring that the sample includes a diverse array of individuals or data points in proportions similar to those in the population, which enhances the generalizability of the results.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true parameter of the population (such as the mean) with a specified level of confidence. It reflects both the variability of the data and the sample size, providing an estimate of the uncertainty associated with the sample statistic.

Margin of Error

The margin of error quantifies the maximum expected difference between the true population parameter and a sample estimate, given a certain level of confidence. It is influenced by the variability within the population, the sample size, and the critical value from the relevant statistical distribution, and it determines the width of the confidence interval.

Random Sampling

Random sampling is a method where each member of a population has an equal chance of being selected, which helps to avoid bias in the sample. This process is essential for ensuring that the sample is representative of the population, thereby increasing the validity and reliability of the statistical inferences made from the sample data.

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