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Sagot :
To test the claim made by Trydint bubble-gum company at the 95% confidence level, we follow steps in hypothesis testing for a population proportion.
### Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p = 0.4 \)[/tex] (The proportion of all people that prefer Trydint gum is 0.4)
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( p \neq 0.4 \)[/tex] (The proportion of all people that prefer Trydint gum is not 0.4)
### Given Data:
- Sample size ([tex]\( n \)[/tex]): 130
- Number of people preferring Trydint ([tex]\( x \)[/tex]): 33
### Step-by-Step Solution:
1. Calculate the Point Estimate:
The point estimate of the population proportion is the sample proportion [tex]\( \hat{p} \)[/tex]:
[tex]\[ \hat{p} = \frac{x}{n} = \frac{33}{130} \approx 0.254 \][/tex]
2. Calculate the Standard Error:
The standard error (SE) for the proportion is given by:
[tex]\[ SE = \sqrt{\frac{p(1 - p)}{n}} \][/tex]
Here, [tex]\( p \)[/tex] is the claimed proportion (0.4), and [tex]\( n \)[/tex] is the sample size (130).
3. Determine the Critical Value:
For a 95% confidence level, the critical value (z-score) for a two-tailed test is approximately 1.96.
4. Calculate the Margin of Error:
The margin of error (ME) is calculated as:
[tex]\[ ME = z \times SE \][/tex]
5. Determine the Confidence Interval:
The 95% confidence interval (CI) for the proportion is given by:
[tex]\[ \hat{p} \pm ME \][/tex]
This means the confidence interval ranges from:
[tex]\[ \text{Lower limit} = \hat{p} - ME \][/tex]
to
[tex]\[ \text{Upper limit} = \hat{p} + ME \][/tex]
### Results:
- Point Estimate:
[tex]\[ \hat{p} = 0.254 \][/tex]
- 95% Confidence Interval:
[tex]\[ \text{Lower limit} = 0.170 \][/tex]
[tex]\[ \text{Upper limit} = 0.338 \][/tex]
### Final Answers:
The point estimate is: [tex]\( 0.254 \)[/tex] (to 3 decimal places).
The 95% confidence interval is: [tex]\( 0.170 \)[/tex] to [tex]\( 0.338 \)[/tex] (to 3 decimal places).
This means we can be 95% confident that the true proportion of people who prefer Trydint gum lies between 0.170 and 0.338. If the claimed proportion 0.4 falls outside this interval, we would have evidence to reject the null hypothesis.
### Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p = 0.4 \)[/tex] (The proportion of all people that prefer Trydint gum is 0.4)
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( p \neq 0.4 \)[/tex] (The proportion of all people that prefer Trydint gum is not 0.4)
### Given Data:
- Sample size ([tex]\( n \)[/tex]): 130
- Number of people preferring Trydint ([tex]\( x \)[/tex]): 33
### Step-by-Step Solution:
1. Calculate the Point Estimate:
The point estimate of the population proportion is the sample proportion [tex]\( \hat{p} \)[/tex]:
[tex]\[ \hat{p} = \frac{x}{n} = \frac{33}{130} \approx 0.254 \][/tex]
2. Calculate the Standard Error:
The standard error (SE) for the proportion is given by:
[tex]\[ SE = \sqrt{\frac{p(1 - p)}{n}} \][/tex]
Here, [tex]\( p \)[/tex] is the claimed proportion (0.4), and [tex]\( n \)[/tex] is the sample size (130).
3. Determine the Critical Value:
For a 95% confidence level, the critical value (z-score) for a two-tailed test is approximately 1.96.
4. Calculate the Margin of Error:
The margin of error (ME) is calculated as:
[tex]\[ ME = z \times SE \][/tex]
5. Determine the Confidence Interval:
The 95% confidence interval (CI) for the proportion is given by:
[tex]\[ \hat{p} \pm ME \][/tex]
This means the confidence interval ranges from:
[tex]\[ \text{Lower limit} = \hat{p} - ME \][/tex]
to
[tex]\[ \text{Upper limit} = \hat{p} + ME \][/tex]
### Results:
- Point Estimate:
[tex]\[ \hat{p} = 0.254 \][/tex]
- 95% Confidence Interval:
[tex]\[ \text{Lower limit} = 0.170 \][/tex]
[tex]\[ \text{Upper limit} = 0.338 \][/tex]
### Final Answers:
The point estimate is: [tex]\( 0.254 \)[/tex] (to 3 decimal places).
The 95% confidence interval is: [tex]\( 0.170 \)[/tex] to [tex]\( 0.338 \)[/tex] (to 3 decimal places).
This means we can be 95% confident that the true proportion of people who prefer Trydint gum lies between 0.170 and 0.338. If the claimed proportion 0.4 falls outside this interval, we would have evidence to reject the null hypothesis.
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