In a recent poll of 1499 random adult Americans, 51% of those surveyed like hot

Question

In a recent poll of 1499 random adult Americans, 51% of those surveyed like hot sauce 1. Find a point estimate for the proportion of all adult Americans who like hot sauce 2. Verify that the conditions are met to construct an interval of confidence for the American adults who like hot sauce Calculate the critical Z value to construct a 90% confidence interval (round to 3 decimals) 3. 4. Calculate the standard of error (round to 4 decimals) Calculate the margin of error to construct the 90% confidence interval (round to 4 decimals) 5. Construct a 92 % confidence interval of all American adults who like hot sauce (round to 4 decimals) 6. 7. Interpret the interval of confidencel

Explanation / Answer

Here sample size = 1499

We find that there are 51% surveryed people like hot sauce.

(1) Point estimate of proportion of people who like hot sauce p^ = 0.51

(2) The conditions and assumptions are:

When constructing confidence intervals the assumptions and conditions of the central limit theorem must be met in order to use the normal model.

Randomization Condition: The data must be sampled randomly. Is one of the good sampling methodologies discussed in the Sampling and Data chapter being used?

Independence Assumption: The sample values must be independent of each other. This means that the occurrence of one event has no influence on the next event. Usually, if we know that people or items were selected randomly we can assume that the independence assumption is met.

10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population.HEre US population is very large so yes it is met.

Sample Size Condition: The sample size must be sufficiently large. Although the Central Limit Theorem tells us that we can use a Normal model to think about the behavior of sample means when the sample size is large enough, it does not tell us how large that should be. If the population is very skewed, you will need a pretty large sample size to use the CLT, however if the population is unimodal and symmetric, even small samples are ok. So think about your sample size in terms of what you know about the population and decide whether the sample is large enough. In general a sample size of 30 is considered sufficient.

3. ere to construct a 90% confidence interval . the critical Z- valueis

Z(critcical) = +-1.645

4. Here standard error se0 = sqrt [p^ (1-p^)/N] = sqrt [0.51 * 0.49/1499] = 0.0129

5 Margin of error for 90% confidence interval = Critical test statistic * standard error

= 1.645 * 0.0129 = 0.02124

6 . 90% confidence interval = p^ +- Margin of error

= 0.51 +- 0.02124 = (0.4888, 0.5312)

7. Here 90% confidence interval means that if we take a sample size of 1499 people then there would be 90% probability that population mean of American people who like hot sauce would be in between these two values.

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