A pharmaceutical company claims that a certain drug causes 80% of people to recover from a disease. Out of 200 people who were given this drug, 153 recovered. For the test of the company’s claim for α = 0.05, do we reject H0 and what is the critical value:
To test the claim of the pharmaceutical company, we will use a hypothesis test.
Null Hypothesis (H0): The drug does not cause 80% of people to recover from the disease.
Alternative Hypothesis (H1): The drug causes 80% of people to recover from the disease.
We will use a significance level (α) of 0.05.
To determine if we reject or fail to reject the null hypothesis, we will use a test statistic and compare it to the critical value.
The test statistic used in this case is the chi-square test statistic. However, since the sample size is greater than 30 and the observations are independent, we can approximate the chi-square distribution with a normal distribution.
First, we calculate the expected number of people who would recover under the assumption that the drug does cause 80% of people to recover.
Expected number of people who would recover = 0.8 * 200 = 160
Now, we can calculate the test statistic:
Test statistic = (Observed - Expected) / sqrt(Expected)
= (153 - 160) / sqrt(160)
≈ -0.438
Next, we compare the test statistic to the critical value.
The critical value is obtained from the standard normal distribution for a two-tailed test and a significance level of 0.05. This critical value corresponds to a z-score of ±1.96.
Since the test statistic (-0.438) is not less than -1.96, and it is not greater than 1.96, we fail to reject the null hypothesis.
Therefore, we do not reject H0, and there is not enough evidence to support the pharmaceutical company's claim that the drug causes 80% of people to recover from the disease.
Note: It's important to note that this analysis assumes that the 200 people were randomly selected and that the sample is representative of the population.
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To determine whether we reject the null hypothesis (H0) and find the critical value, we can perform a hypothesis test using the given information.
H0 (null hypothesis): The drug does not cause people to recover from the disease (p = 0.80)
H1 (alternative hypothesis): The drug does cause people to recover from the disease (p > 0.80)
To conduct the hypothesis test, we will perform a one-sample proportion test:
Step 1: Calculate the sample proportion
The sample proportion (p̂) is calculated by dividing the number of people who recovered (153) by the total sample size (200):
p̂ = 153/200 = 0.765
Step 2: Calculate the test statistic
The test statistic (Z) is calculated using the formula:
Z = (p̂ - p) / √[(p * (1 - p)) / n]
Where:
p̂ = sample proportion
p = hypothesized population proportion (null hypothesis)
n = sample size
In this case, p = 0.80 and n = 200. Substituting these values into the formula:
Z = (0.765 - 0.80) / √[(0.80 * (1 - 0.80)) / 200]
Z = -0.035 / 0.016
Step 3: Find the critical value
The critical value is the value that separates the regions of acceptance and rejection in a hypothesis test. In this case, we are conducting a one-tailed test with α = 0.05 significance level.
To find the critical value, we need to use a standard normal distribution table or a calculator. From the table or calculator, we find that the critical value for α = 0.05 is approximately 1.645.
Since we are performing a one-tailed test and the alternative hypothesis is p > 0.80, we reject the null hypothesis if the test statistic (Z) is greater than the critical value (1.645).
Step 4: Compare the test statistic with the critical value
The test statistic (Z) calculated earlier is -0.035 / 0.016, which is less than the critical value of 1.645.
Since the test statistic is not greater than the critical value, we fail to reject the null hypothesis.
Therefore, based on the given information and a significance level (α) of 0.05, we do not reject the null hypothesis (H0), and the critical value is 1.645.
To determine if we reject the null hypothesis (H0) and find the critical value, we need to perform a hypothesis test.
H0: The drug does not cause 80% of people to recover from the disease.
H1: The drug causes 80% of people to recover from the disease.
We can use a binomial proportion test to conduct this hypothesis test.
1. Calculate the sample proportion:
Sample proportion (p̂) = Number of successes (recovered) / Sample size
p̂ = 153 / 200 = 0.765
2. Calculate the standard error:
Standard Error (SE) = sqrt( (p̂ * (1 - p̂)) / n )
SE = sqrt( (0.765 * 0.235) / 200 ) = 0.025
3. Calculate the test statistic:
Test Statistic (z) = (observed - expected) / SE
Here, the expected proportion is 80% or 0.80.
z = (0.765 - 0.80) / 0.025 = -1.4
4. Determine the critical value:
The critical value depends on the significance level (α) chosen for the test.
Given α = 0.05, we need to find the critical value for a two-tailed test.
For a two-tailed test at α = 0.05, the critical value is ±1.96.
5. Decide whether to reject or fail to reject the null hypothesis:
If the test statistic falls within the rejection region (outside the critical value range), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the test statistic of -1.4 falls within the range of ±1.96, so we fail to reject the null hypothesis. There is not enough evidence to support the claim that the drug causes 80% of people to recover from the disease.