Question

The average weekly earnings in the leisure and hospitality industry group for a re‐ cent year was $273. A random sample of 40 workers showed weekly average ear‐ nings of $285 with the population standard deviation equal to 58. At the 0.05 level of significance can it be concluded that the mean differs from $273? Find a 95% con‐ fidence interval for the weekly earnings and show that it supports the results of the hypothesis test.

175

likes
877 views

Answer to a math question The average weekly earnings in the leisure and hospitality industry group for a re‐ cent year was $273. A random sample of 40 workers showed weekly average ear‐ nings of $285 with the population standard deviation equal to 58. At the 0.05 level of significance can it be concluded that the mean differs from $273? Find a 95% con‐ fidence interval for the weekly earnings and show that it supports the results of the hypothesis test.

Expert avatar
Hermann
4.6
118 Answers
To determine whether the mean weekly earnings differs from $273, we will conduct a one-sample z-test.

Let's denote the population mean as μ, the sample mean as x̄, the population standard deviation as σ, the sample size as n, and the level of significance as α.

Given:
x̄ = $285
σ = $58
n = 40
α = 0.05

Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
H0: The mean weekly earnings, μ, is equal to $273.
Ha: The mean weekly earnings, μ, differs from $273.

Step 2: Calculate the test statistic.
We will use the formula for the z-test statistic:
z = \frac{x̄ - μ}{\frac{σ}{\sqrt{n}}}

Substituting in the given values:
z = \frac{285 - 273}{\frac{58}{\sqrt{40}}}

Step 3: Determine the critical value.
Since we are conducting a two-tailed test at the 0.05 level of significance, we need to find the critical z-value for α/2 = 0.025. This value can be obtained from the standard normal distribution table or calculator.

The critical z-value for a 0.025 level of significance is approximately ±1.96.

Step 4: Make a decision.
If the test statistic z falls outside the critical region (greater than 1.96 or less than -1.96), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 5: Calculate the p-value.
The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.

To calculate the p-value, we can use a standard normal distribution table or a calculator. The p-value is the area under the standard normal curve that corresponds to the absolute value of the test statistic.

Step 6: Determine the conclusion.
If the p-value is less than the level of significance (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Now let's perform the calculations:

z = \frac{285 - 273}{\frac{58}{\sqrt{40}}} = \frac{12}{9.176 } ≈ 1.31

The critical z-value at the 0.025 level of significance is ±1.96.

Since the test statistic z (1.31) falls within the range -1.96 to 1.96, we fail to reject the null hypothesis.

Now, let's proceed to find the 95% confidence interval for the weekly earnings.

The formula for the confidence interval is:

\text{Confidence Interval}=\mu_0\pm z\frac{σ}{\sqrt{n}}

Plugging in the given values:

\text{Confidence Interval}=273\pm1.96\frac{58}{\sqrt{40}}

Calculating this, we get:

\text{Confidence Interval}=273\pm17.974

Thus, the 95% confidence interval for the weekly earnings is [255.026 : 290.974]. The population mean ($273) is within the confidence interval. Therefore, we cannot reject the null hypothesis.

Answer: No, at the 0.05 level of significance, it cannot be concluded that the mean weekly earnings differ from $273.

Frequently asked questions (FAQs)
What is the mode of the following numbers: 5, 7, 7, 9, 12, 12, 12, 16, 19?
+
Question: In a circle, if angle ACB is 70°, and angle ABC is twice the size of angle BCA, what is the measure of angle BCA?
+
What is the formula for the surface area of a cone?
+
New questions in Mathematics
5 . {2/5 + [ (8/-9) - (1/-7) + (-2/5) ] ÷ (2/-5)} . 8/15
calculate the following vector based on its base vectors a= -18i,26j
Since one of the three integers whose product is (-60) is (+4), write the values that two integers can take.
Use the digits of 1,9,2,3 to come up with all the numbers 98 and 95
For a temperature range between -3 degrees Celsius to 5 degrees Celsius, what is the temperature range in degrees Farenheight
If L (-2, -5) reflected across y = -4. What are the coordinates of L?
5 people can complete a task in 72 hours. How many people are needed to complete the task in 60 hours.
Pedro had 80% of the amount needed to buy a game. Of this amount, you spent 15% on a watch and therefore, you will need to add another R$640.00 to purchase this game. Is the value of the game?
According to a survey in a country 27% of adults do not own a credit card suppose a simple random sample of 800 adults is obtained . Describe the sampling distribution of P hat , the sample proportion of adults who do not own a credit card
User Before the election, a poll of 60 voters found the proportion who support the Green candidate to be 25%. Calculate the 90% confidence interval for the population parameter. (Give your answers as a PERCENTAGE rounded to TWO DECIMAL PLACES: exclude any trailing zeros and DO NOT INSERT THE % SIGN) Give the lower limit of the 90% confidence interval Give the upper limit of the 90% confidence interval
3/9*4/8=
In an audience of 4000 people, 2 people are chosen, at random, to appear on stage. How many ways can the people be chosen?
Solve equations by equalization method X-8=-2y 2x+y=7
A bag has 4 green lollipops, 3 white lollipops, and 1 black lollipop. What is the probability of drawing a white lollipop?
A company has had the following data for two consecutive years. Total, asset item 3,100,500 euros 3,300,550 euros. Net amount of business figures 4,755,250 euros /5,100 euros Average number of workers employed during the year 64/70 You can present a balance sheet in an abbreviated form
2.3 X 0.8
did an analysis of dropout from the nursing faculty at the Universidad Veracruzana. With a poblation of 122 students, it turned out that according to the gender data, the female sex predominates with 82%, and the male sex male is found with 12%. The main factors why students drop out are, first of all, "Not "re-enrolled" at 49%, second place "Personal reasons" at 20%, third place "change of school" in 11%, "lack of documents" and "economic reasons" in 7%, change of residence and lack of social service in 3%. Of this sample, how many students dropped out for other reasons?
question 1 Consider a sample space S, and two events A and B such that P(A ∩ B) = 0.2, P(A ∪ B) = 0.6, P(B ∪ ̄A) = 0.8 (a) [0.5 points] Calculate P (A). (b) [0.5 points] Calculate P (B)
A rectangular swimming pool has a length of 14 feet, a width of 26 feet and a depth of 5 feet. Round answers to the nearest hundredth as needed. (a) How many cubic feet of water can the pool hold? cubic feet (b) The manufacturer suggests filling the pool to 95% capacity. How many cubic feet of water is this? cubic feet
The car with an irresponsible driver starts to brake when it goes through a red light. When passing the traffic light, he does so at a speed of 115 kph in the right lane. Further ahead, 70 meters from the traffic light, a child is crossing the street and falls. If the effect of the car's brakes is equivalent to a deceleration of magnitude 5.7m/s². Is the child hit by the car or not? How far from the traffic light does the car stop?