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A statistics instructor randomly selected four bags of oranges, each bag labeled 10 pounds, and weighed the bags. They weighed \(10.2,10.5,10.3\), and \(10.3\) pounds. Assume that the distribution of weights is Normal. Find a \(95 \%\) confidence interval for the mean weight of all bags of oranges. Use technology for your calculations. a. Decide whether each of the following three statements is a correctly worded interpretation of the confidence interval, and fill in the blanks for the correct option(s). i. I am \(95 \%\) confident that the population mean is between ____. ii. There is a \(95 \%\) chance that all intervals will be between ____. and ____. iii. I am \(95 \%\) confident that the sample mean is between____. and ____. b. Does the interval capture 10 pounds? Is there enough evidence to reject the null hypothesis that the population mean weight is 10 pounds? Explain your answer.

Step by step solution

01

Find the Sample Mean and Standard Deviation

Start by finding the sample mean and sample standard deviation. The sample mean \(\bar{x}\) is the sum of the sample weights divided by the number of samples. The sample standard deviation \(s\) can be found using a calculator or statistics software.

02

Determine the Confidence Interval

The 95% confidence interval for a normal distribution is the sample mean ± 1.96 times the standard error. The standard error is the standard deviation divided by the square root of the sample size.

03

Interpret the Confidence Interval

The correct interpretation of the confidence interval is given by statement i. The phrase, 'I am 95% confident that the population mean is between ___' suggests that if the procedure was repeated many times, 95% of the resulting intervals would contain the population mean.

04

Test the Null Hypothesis

The null hypothesis states that the population mean weight is 10 pounds. If the confidence interval calculated includes 10, then there isn't enough evidence to reject the null hypothesis, otherwise it should be rejected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean

In statistics, the sample mean is a central concept. It's the average of the values you collect. When collecting data, like the weights of orange bags, you sum these values and divide by the number of observations. For example, if you have weights of 10.2, 10.5, 10.3, and 10.3 pounds, the sample mean would be calculated as follows:\[\bar{x} = \frac{10.2 + 10.5 + 10.3 + 10.3}{4} = 10.325\]This figure provides a basic estimate of the central tendency of the sample data. It helps us understand the typical value within our collected data set.
The sample mean is crucial in forming the basis for further statistical analysis, like creating confidence intervals or conducting hypothesis tests.

Standard Deviation

Standard deviation is a measure that tells us how much the values in our data set differ from the average value, or mean. In simple terms, it shows us the extent of variation or dispersion of our data points.
To calculate the standard deviation (\[s\]) for the weights of oranges, you first need the variance, which is the average squared deviation from the mean. Using the sample mean previously calculated, subtract the mean from each weight, square the result, and take the average of those squared differences:

• Calculate difference from the mean for each observation.
• Square these differences.
• Find the average of those squared differences.

The standard deviation is useful because it provides insights into the spread of the distribution, indicating how tightly or loosely the data is clustered around the mean.

Null Hypothesis

In hypothesis testing, the null hypothesis is a default or original assumption that there is no effect or no difference. It is a starting point that we test to find evidence against it.
For the exercise, the null hypothesis (\[H_0\]) states that the population mean weight of the orange bags is 10 pounds. This assumption acts as a baseline that researchers will either reject or fail to reject based on the data collected.
Testing the null hypothesis involves statistical calculations such as computing confidence intervals or other metrics, which determine whether the observed data provides enough evidence to conclude that the null hypothesis isn’t true.

Normal Distribution

The normal distribution, commonly known as the bell curve, is a fundamental concept in statistics. It depicts data that clusters around a mean or average, with data points decreasing in frequency as they move away from the mean.
Its shape is symmetrical, with a peak at the mean, and it always follows a specific mathematical function. Many natural phenomena, including weights of objects, often approximate this pattern.

• The mean, median, and mode of a normal distribution are all the same.
• It is defined by its mean and standard deviation.
• Approximately 68% of data falls within one standard deviation of the mean.

Understanding normal distribution helps us make inferences about a population from a sample, aiding the construction of confidence intervals.