MAT-243 - 3-3 Discussion- Confidence Intervals and Hypothesis Testing

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Jan 9, 2024

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3-3 Discussion: Confidence Intervals and Hypothesis Testing In your initial post, address the following items. Be sure to answer the questions about both confidence intervals and hypothesis testing. 1. In the Python script, you calculated the sample data to construct a 90% and 99% confidence interval for the average diameter of ball bearings produced from this manufacturing process. These confidence intervals were created using the Normal distribution based on the assumption that the population standard deviation is known and the sample size is sufficiently large. Report these confidence intervals rounded to two decimal places. See Step 2 in the Python script. 2. Interpret both confidence intervals. Make sure to be detailed and precise in your interpretation. It has been claimed from previous studies that the average diameter of ball bearings from this manufacturing process is 2.30 cm. Based on the sample of 50 that you collected, is there evidence to suggest that the average diameter is greater than 2.30 cm? Perform a hypothesis test for the population mean at alpha = 0.01. In your initial post, address the following items: 1. Define the null and alternative hypothesis for this test in mathematical terms and in words. 2. Report the level of significance. 3. Include the test statistic and the P-value. See Step 3 in the Python script. (Note that Python methods return two tailed P-values. You must report the correct P-value based on the alternative hypothesis.) 4. Provide your conclusion and interpretation of the results. Should the null hypothesis be rejected? Why or why not? Hello All, 1. The 90 % confidence intervals rounded to two decimal places in the sample data is (2.42 , 2.65). This means that the average diameter of ball bearings is between 2.42 and 2.65 .

The 99% confidence intervals rounded to two decimal places in the sample data is (2.35, 2.72). This shows that 99% confidence intervals have an average diameter of ball bearings between 2.35 and 2.72 2. Confidence interval is combined with a probability statement. The 90 % confidence interval of (2.35, 2.65) implies that if samples are drawn repeatedly, the probability that the true average diameter of the ball bearings is in the interval of 90%. The 99 % confidence interval exists that the probability of the true average diameter of the ball bearings is in the range between 2.35 and 2.72. It has been claimed from previous studies that the average diameter of ball bearings from this manufacturing process is 2.30 cm. Based on the sample of 50 that you collected, is there evidence to suggest that the average diameter is greater than 2.30 cm? Perform a hypothesis test for the population mean at alpha = 0.01. 1. The null hypothesis denoted H 0 is defined as the statement assumed to be true unless sufficient data indicates otherwise. Therefore, the null hypothesis for this previous study is that the average diameter of the ball bearing from this manufacturing process is H 0 : μ = 2.30 cm. While the alternative hypothesis denoted H a, is the opposite of the null hypothesis. As a result, the alternative hypothesis is that the average diameter of the ball bearing from this manufacturing process is H a : μ > 2.30 cm and is considered a right-tailed alternative hypothesis because the value of a parameter is greater than the value asserted in the null hypothesis. 2. The level of significance is 0.01 which means that there are 1% chances that the average diameter of the ball bearing is greater than 2.30 cm. 3. My Z- test - Statistic is 4.02. The right tailed p -value is 0.000029 or 0.0001 4. Since the p-value (0.001) is lesser than the significance level α = 0.01, sufficient statistical evidence exists to reject the null hypothesis that the that the average diameter of the ball bearing from this manufacturing process is H 0 : μ = 2.30 cm. In your follow-up posts to other students, review your peers' calculations and provide some analysis and interpretation:

1. How do their confidence intervals compare with yours? 2. If the population standard deviation is unknown and the sample size is not sufficiently large, would you still use the Normal distribution to calculate these confidence intervals, or would you choose another distribution? If the latter, which distribution would you choose? Hello, Your rounded 99% confidence interval and 90% confidence interval are very similar to mine. My 90% confidence interval is (2.35, 2.65), whereas yours is (2.43, 2.66). My 99% confidence interval is (2.35, 2.65), whereas yours is (2.36, 2.72). Although your test-statistic result is 3.46 and mine is 4.02, the null hypothesis is rejected because our p-values are both smaller than the alpha value (0.01) in both cases. If the population standard deviation is unknown and the sample size is not sufficiently large, I would use the Student t-distribution because according to the condition to use the student t- distribution : In most cases, the normality of the underlying distribution cannot be determined. Thus, the sample size determines whether the t -distribution can be used. Hello, Because yours is (2.30, 2.53) and mine is (2.35, 2.65), our 90% confidence levels are rather close to one another. Since my 99% is (2.36, 2.72) and yours is (2.23, 2.60), your 99% level is slightly higher than mine. There is insufficient evidence to reject the null hypothesis because your p- value is greater than your significance level. If the population standard deviation is unknown and the sample size is not sufficiently large, I would use the Student t-distribution because according to the condition to use the student t- distribution : In most cases, the normality of the underlying distribution cannot be determined. Thus, the sample size determines whether the t -distribution can be used. This post shows good progress towards proficiency, and there are some things to address as well. Your hypotheses for your hypothesis test are correctly set up, which is excellent and is one of the most important things to master for hypothesis testing. One thing you really want to address is to begin each discussion post with an introduction to the overall context of the problem. You want your reader to understand what it is that you will be analyzing, in this case the mean diameter size for ball bearings from a certain factory. Your discussion of confidence intervals is not very clear. The correct interpretation of the confidence intervals is that, for the 90% interval for example, there is a 90% chance that the true population mean, in other words the actual average diameter of bearings produced at the factory, is within the specified interval. Your interpretation of the meaning of alpha is not quite correct. This can be a tricky one sometimes. The alpha level is the maximum chance we will allow of a Type 1 error, an error in which the null hypothesis is actually true but we mistakenly reject it. The p-value gives the calculated probability of getting the results that we got if the null hypothesis were true. That is why as long as the p-value is below alpha, we have less than 1% (for the example of a 1% alpha value) chance of mistakenly rejecting an actually true null hypothesis, since in that case there

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will be less than 1% chance that the null is true. Note that the alpha value does not tell us the likelihood of the alternate hypothesis being true. The p-value tells us the likelihood of getting the results we got under the assumption that the null hypothesis is true. So in your case, there was much less than a 1% chance that you would get the results that you got if the true mean diameter were actually equal to 2.30cm. Since the chance of getting results that you got was so low, lower than 1%, we conclude that the null hypothesis must not be true, and we instead assume that the alternate hypothesis must be correct. Does that make sense? It often takes students a few times to really get the hang of hypothesis testing, so hang in there. You nailed defining the hypotheses. Review the announcement I posted on hypothesis testing and post any questions in the General Questions discussion board.

MAT-243 - 3-3 Discussion- Confidence Intervals and Hypothesis Testing

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