Homework 6
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University of New Mexico, Main Campus *
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1350
Subject
Mathematics
Date
Jan 9, 2024
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docx
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3
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Alysandra Dominguez
MATH1350
HOMEWORK 6
1a. A confidence interval is a statistical term that estimates the range of values that a population parameter, such the population mean or percentage, is likely to fall within in inferential statistics.
1b. Confidence intervals offer a framework for estimating population parameter values and drawing conclusions about them while expressing the degree of confidence attached to those values. They are an important statistical tool that help researchers come to ethical and rigorous findings, compare ideas, and guide decision-making.
2. Sample size, selected confidence level, data variability, standard error, population size, sampling strategy, estimator selection, and outlier presence all have an impact on the size of a
confidence interval. When analyzing and applying confidence intervals in statistical analysis, it is essential to comprehend these variables and how they interact. To derive relevant findings and generate appropriate inferences from their data, researchers must carefully analyze these elements.
3a. A lower p-value denotes more robust evidence against the null hypothesis. An extremely tiny p-value (usually less than a selected significance threshold, commonly represented by the letter alpha, like 0.05) suggests that the observed data are unlikely to have happened in the event that the null hypothesis were correct. This might result in the alternative hypothesis being accepted and the null hypothesis being rejected.
3b. p ≤ 0.01: Strong evidence against the null hypothesis. 0.01 < p ≤ 0.05: Moderate evidence against the null hypothesis.
0.05 < p ≤ 0.10: Weak evidence against the null hypothesis. p > 0.10: Weak evidence against the null hypothesis.
3c. Rejecting the null hypothesis becomes more challenging when the significance threshold is lowered to 0.01. It indicates that further proof supporting the alternative theory is required. Consequently, you will be less likely to reject the null hypothesis and less likely to commit a Type
I mistake, or false positive. You run the danger of committing a Type II mistake, or false negative,
though, which is when you fail to rule out the null hypothesis when it is incorrect.
4a. Population: Hotel managers in Chicago and Detroit, particularly those in charge of establishments with 200–500 rooms, make up the study's population.
Major Shortcomings: The survey was completed by 101 managers out of 400, which is a relatively low response rate. Additionally, there may be non-response bias, meaning that the satisfaction levels of non-respondents may differ from those of respondents. Lastly, because the focus of the data was on a specific range of hotel sizes, there may be a potential lack of representativeness of all hotel managers in the two cities.
4b. Sample Mean (
) = 5.396
x̄
Sample Standard Deviation (s) = 1.75
Sample Size (n) = 101
Confidence Level (1 - α) = 95%
Z-Score for a 95% confidence interval ≈ 1.96 (obtained from standard normal distribution table)
ME = (Z-Score) * (s / √n) = 1.96 * (1.75 / √101) ≈ 0.308
Confidence Interval:
95% Confidence Interval ≈ (
- ME, + ME)
x̄
x̄
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95% Confidence Interval ≈ (5.396 - 0.308, 5.396 + 0.308)
95% Confidence Interval ≈ (5.088, 5.704)
4c. The population mean satisfaction score for computer system usability among hotel managers in Chicago and Detroit (hotels with 200–500 rooms) is between 5.088 and 5.704, with a 95% confidence interval.
4d. Sample Mean (
) = 4.398
x̄
Sample Standard Deviation (s) = 1.75
Sample Size (n) = 101
Confidence Level (1 - α) = 99%
Z-Score for a 99% confidence interval ≈ 2.58 (obtained from standard normal distribution table)
ME = (Z-Score) * (s / √n) = 2.58 * (1.75 / √101) ≈ 0.460
Confidence Interval:
99% Confidence Interval ≈ (
- ME, + ME)
x̄
x̄
99% Confidence Interval ≈ (4.398 - 0.460, 4.398 + 0.460)
99% Confidence Interval ≈ (3.938, 4.858)
4e. A 99% confidence level indicates that the population mean satisfaction score for computer training among hotel managers in 200–500 room hotels in Chicago and Detroit is between 3.938 and 4.858. This greater degree of confidence is reflected in the broader confidence interval, which
is less accurate than the 95% confidence interval in Part B.
5. Nicotine mg. Mean
1.64166667
95% Confidence Level
Standard Erro
0.08317013
Median
1.75
Lower Limit
1.46961614
Mode
1.9
Standard Dev
0.40744876
Upper Limit
1.81371719
Sample Varia
0.16601449
Kurtosis
-0.7402907
Interpretation:
Skewness
-0.177423
We are 95% confident the true Nictone Range
1.5
(N=24)
Minimum
0.9
Maximum
2.4
Sum
39.4
Count
24
Confidence Le
0.17205052
5a. The confidence interval is: (1.469, 1.813).
5b. If repeated samples were taken and the 95% confidence interval computed for each sample, 95% of the intervals would contain the population mean.
5c. No, the interval does not contain the suggested nicotine mg by the maker. So the sample is not
a good representative of the population.
6. The poll's results for males have a higher margin of error than those for women, partly because the sample size for men is lower. To get more accurate predictions with reduced error margins, a
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large enough sample size is necessary. The real population parameter is 95% likely to fall inside the given margin of error, according to the 95% confidence level.
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