Can you explain what it means to be 95 percent …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

The manufacturer of a certain type of new cell phone battery claims that the average life span of the batteries is 500 charges; that is, the battery can be charged at least 500 times before failing. To investigate the claim, a consumer group will select a random sample of cell phones with the new battery and use the phones through 500 charges of the battery. The proportion of batteries that fail to last through 500 charges will be recorded. The results will be used to construct a 95 percent confidence interval to estimate the proportion of all such batteries that fail to last through 500 charges. (a) Explain in context what it means to be 95 percent confident. (b) Suppose the consumer group conducts its investigation with a random sample of 5 cell phones with the new battery, and 1 battery out of the 5 fails to last through 500 charges. Verify all conditions for inference for a 95 percent confidence interval for a population proportion. Indicate whether any condition has not been met. Do not construct the interval. In some studies, large sample sizes are not feasible because of the time and expense involved. However, small sample sizes can create issues with inference procedures. A simulation can be used to investigate what might happen with intervals constructed from small sample sizes. Consider a population in which 30 percent of the population displays a certain characteristic. For each trial of the simulation, 5 observations are selected from the population and the sample proportion p√ɬå√¢¬Ä¬ö is calculated, where p√ɬå√¢¬Ä¬ö represents the proportion in the sample that display the characteristic. The following table shows the frequency distribution of p√ɬå√¢¬Ä¬ö in the 1,000 trials. Also shown are the upper and lower endpoints of a 95 percent confidence created from the value of p√ɬå√¢¬Ä¬ö, using the formula p√ɬå√¢¬Ä¬ö√Ǭ±1.96p√ɬå√¢¬Ä¬ö(1-p√ɬå√¢¬Ä¬ö)n—–√¢¬à¬ö. For example, the sample proportion of 0.4 occurred 309 times in the 1,000 trials and produced a confidence interval of (-0.029,0.829). p√ɬå√¢¬Ä¬ö Frequency Lower Endpoint Upper Endpoint 0 168 0 0 0.2 360 -0.151 0.551 0.4 309 -0.029 0.829 0.6 133 0.171 1.029 0.8 28 0.449 1.151 1.0 2 1 1 (c) Based on the simulation, What proportion of the 95 percent confidence intervals capture the population proportion of 0.3 ? Explain how you determined your answer. (d) For small sample sizes, an alternate method of constructing a confidence interval is available. To implement this method, first, include an additional 4 observations in the sample, 2 of which are successes and 2 of which are failures. Second, calculate a new sample proportion p√ɬå√¢¬Ä¬önew for the new sample. Finally, construct a confidence interval using p√ɬå√¢¬Ä¬önew, with the formula p√ɬå√¢¬Ä¬önew√Ǭ±1.96p√ɬå√¢¬Ä¬önew(1-p√ɬå√¢¬Ä¬önew)n+4———–√¢¬à¬ö. (i) For the cell phone batteries, consider a sample of 5 in which 1 battery fails to last through 500 charges. Using the alternate method described, What is the value of p√ɬå√¢¬Ä¬önew? Show your work. The following table shows the results of the original simulation with revised 95 percent confidence intervals constructed with the alternate method. Original p√ɬå√¢¬Ä¬ö Frequency Revised Lower Endpoint Revised Upper Endpoint 0 168 -0.049 0.494 0.2 360 0.025 0.641 0.4 309 0.120 0.769 0.6 133 0.231 0.880 0.8 28 0.359 0.975 1.0 2 0.506 1.049 (ii) Based on the results of the simulation, is the alternate method better than the traditional method described in part (b) to construct a 95 percent confidence interval with a small sample size? Explain your reasoning.

Short Answer

To ensure a 95% confidence interval for a population proportion, it is essential to validate conditions such as randomness, normality (by checking if both np and n(1-p) are greater than 10), and independence (sample size should be less than 10% of the population). Additionally, an alternate method involves adding 2 successes and 2 failures to small sample sizes to enhance reliability and reduce variability in estimates.

Step-by-Step Solution

Step 1: Understanding 95% Confidence Level

To be 95% confident means that if you took numerous random samples to create a 95% confidence interval from each, you would expect about 95% of those intervals to include the true population proportion. This implies a strong likelihood of capturing the population parameter, without guaranteeing that any specific interval is correct.

Step 2: Verifying Conditions for Confidence Interval

Before constructing a 95% confidence interval for a population proportion, you must ensure that certain conditions are met:

  • Randomness: The sample must be selected randomly.
  • Normality: The sample size should be large enough for the sampling distribution to be approximately normal, typically verified with the rule that both np and n(1-p) should be greater than 10.
  • Independence: The sample size must be less than 10% of the population size to ensure independence between observations.

Step 3: Assessing the Alternate Method

The alternate method for constructing confidence intervals can improve estimates for small sample sizes. This method involves adjusting the number of successes and failures based on a fixed addition:

  • Adding 2 successes and 2 failures to the sample size results in a more reliable estimate of the proportion.
  • This revised method yields intervals that are positive and more aligned with typical values, reducing variability and avoiding impractical results.

Related Concepts

Confidence Level

A statistical measure that quantifies the degree of certainty that a population parameter lies within a certain range, typically expressed as a percentage, such as 95%.

Confidence Interval

A range of values derived from sample data that is likely to contain the true population parameter, with a specified level of confidence (e.g., 95%).

Random Sampling

A technique where each individual or item in a population has an equal chance of being selected for the sample, ensuring that the sample accurately represents the population.

Table Of Contents
  1. The manufacturer of a certain type of new cell phone battery claims that the average life span of the batteries is 500 charges; that is, the battery can be charged at least 500 times before failing. To investigate the claim, a consumer group will select a random sample of cell phones with the new battery and use the phones through 500 charges of the battery. The proportion of batteries that fail to last through 500 charges will be recorded. The results will be used to construct a 95 percent confidence interval to estimate the proportion of all such batteries that fail to last through 500 charges. (a) Explain in context what it means to be 95 percent confident. (b) Suppose the consumer group conducts its investigation with a random sample of 5 cell phones with the new battery, and 1 battery out of the 5 fails to last through 500 charges. Verify all conditions for inference for a 95 percent confidence interval for a population proportion. Indicate whether any condition has not been met. Do not construct the interval. In some studies, large sample sizes are not feasible because of the time and expense involved. However, small sample sizes can create issues with inference procedures. A simulation can be used to investigate what might happen with intervals constructed from small sample sizes. Consider a population in which 30 percent of the population displays a certain characteristic. For each trial of the simulation, 5 observations are selected from the population and the sample proportion p√ɬå√¢¬Ä¬ö is calculated, where p√ɬå√¢¬Ä¬ö represents the proportion in the sample that display the characteristic. The following table shows the frequency distribution of p√ɬå√¢¬Ä¬ö in the 1,000 trials. Also shown are the upper and lower endpoints of a 95 percent confidence created from the value of p√ɬå√¢¬Ä¬ö, using the formula p√ɬå√¢¬Ä¬ö√Ǭ±1.96p√ɬå√¢¬Ä¬ö(1-p√ɬå√¢¬Ä¬ö)n—–√¢¬à¬ö. For example, the sample proportion of 0.4 occurred 309 times in the 1,000 trials and produced a confidence interval of (-0.029,0.829). p√ɬå√¢¬Ä¬ö Frequency Lower Endpoint Upper Endpoint 0 168 0 0 0.2 360 -0.151 0.551 0.4 309 -0.029 0.829 0.6 133 0.171 1.029 0.8 28 0.449 1.151 1.0 2 1 1 (c) Based on the simulation, What proportion of the 95 percent confidence intervals capture the population proportion of 0.3 ? Explain how you determined your answer. (d) For small sample sizes, an alternate method of constructing a confidence interval is available. To implement this method, first, include an additional 4 observations in the sample, 2 of which are successes and 2 of which are failures. Second, calculate a new sample proportion p√ɬå√¢¬Ä¬önew for the new sample. Finally, construct a confidence interval using p√ɬå√¢¬Ä¬önew, with the formula p√ɬå√¢¬Ä¬önew√Ǭ±1.96p√ɬå√¢¬Ä¬önew(1-p√ɬå√¢¬Ä¬önew)n+4———–√¢¬à¬ö. (i) For the cell phone batteries, consider a sample of 5 in which 1 battery fails to last through 500 charges. Using the alternate method described, What is the value of p√ɬå√¢¬Ä¬önew? Show your work. The following table shows the results of the original simulation with revised 95 percent confidence intervals constructed with the alternate method. Original p√ɬå√¢¬Ä¬ö Frequency Revised Lower Endpoint Revised Upper Endpoint 0 168 -0.049 0.494 0.2 360 0.025 0.641 0.4 309 0.120 0.769 0.6 133 0.231 0.880 0.8 28 0.359 0.975 1.0 2 0.506 1.049 (ii) Based on the results of the simulation, is the alternate method better than the traditional method described in part (b) to construct a 95 percent confidence interval with a small sample size? Explain your reasoning.

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