The life in hours of a battery is known to be approximately normally distributed, with standard deviation o = 1.25 hours. A random sample of 10 batteries has a mean life of = 40.5 hours. (a) Is there evidence to support the claim that battery life exceeds 40 hours? Use a = 0.020. The battery life is not significantly different greater than 40 hours at a = 0.020. (b) What is the P-value for the test in part (a)? P-value = 0.1030 Round your answer to three decimal places (e.g. 98.765). (c) What is the 3-error for the test in part (a) if the true mean life is 42 hours? 3- i 0.00045 Round your answer to five decimal places (e.g. 98.76543). (d) What sample size would be required to ensure that 3 does not exceed 0.1 if the true mean life is 44 hours? batteries 1
The life in hours of a battery is known to be approximately normally distributed, with standard deviation o = 1.25 hours. A random sample of 10 batteries has a mean life of = 40.5 hours. (a) Is there evidence to support the claim that battery life exceeds 40 hours? Use a = 0.020. The battery life is not significantly different greater than 40 hours at a = 0.020. (b) What is the P-value for the test in part (a)? P-value = 0.1030 Round your answer to three decimal places (e.g. 98.765). (c) What is the 3-error for the test in part (a) if the true mean life is 42 hours? 3- i 0.00045 Round your answer to five decimal places (e.g. 98.76543). (d) What sample size would be required to ensure that 3 does not exceed 0.1 if the true mean life is 44 hours? batteries 1
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9.1.2 8 please help correct part C) and D) both are incorrect
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The life in hours of a battery is known to be approximately normally distributed, with a standard deviation \( \sigma = 1.25 \) hours. A random sample of 10 batteries has a mean life of \( \bar{x} = 40.5 \) hours.
**(a)** Is there evidence to support the claim that battery life exceeds 40 hours? Use \( \alpha = 0.020 \).
The battery life is **not** significantly different greater than 40 hours at \( \alpha = 0.020 \).
**(b)** What is the P-value for the test in part (a)?
\( P \)-value = **0.1030**
Round your answer to three decimal places (e.g. 98.765).
**(c)** What is the \( \beta \)-error for the test in part (a) if the true mean life is 42 hours?
\( \beta = \) **0.00045**
Round your answer to five decimal places (e.g. 98.76543).
**(d)** What sample size would be required to ensure that \( \beta \) does not exceed 0.1 if the true mean life is 44 hours?
Sample size = **1** batteries
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This detailed explanation provides an insight into hypothesis testing involving battery life expectancy. Each step is outlined with specific statistical measures, emphasizing the importance of P-value and β-error in making informed decisions based on sample data.
Transcribed Image Text:The information presented is a statistical analysis regarding the life in hours of a battery, assumed to be normally distributed with a standard deviation of \( \sigma = 1.25 \) hours. A random sample of 10 batteries shows a mean life of \( \bar{x} = 40.5 \) hours.
### Analysis
#### (a) Hypothesis Test
The question is whether there is evidence to support the claim that the battery life exceeds 40 hours at a significance level of \( \alpha = 0.020 \).
- **Conclusion**: The battery life **is not** significantly greater than 40 hours at \( \alpha = 0.020 \).
#### (b) P-value Calculation
- **P-value**: The P-value for this test is calculated to be \( 0.1030 \). This value is rounded to three decimal places (e.g., 98.765).
#### (c) β-error Calculation
- **β-error** (Type II error): If the true mean life is 42 hours, the β-error for the test in part (a) is \( 0.00003 \). This value is presented rounded to five decimal places (e.g., 98.76543).
#### (d) Sample Size Determination
- To ensure that the β-error does not exceed 0.1 if the true mean life is 44 hours, the required sample size is calculated to be **1** battery.
### Additional Resource
The analysis refers to "Statistical Tables and Charts" as a resource for further insights or tables used in the calculations.
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