A survey of 23 customers at a shopping centre asked the distance(km) travelled to reach the shopping centre. The data was entered into Minitab to determine the descriptive statistics.a survey of 23 customers at a shopping centre asked the distance(km) travelled to reach the shopping centre. The data was entered into Minitab to determine the descriptive statistics.
Variable Distance N=23, Mean=8.221, Median=8.130, StDev=1.467, Variable Distance Minimum=5.405, maximum=11.171, Q1=7.402, Q3=9.293
Calculate the 90% confidence interval for the true mean and provide a concluding statement.
State 2 assumptions necessary to do the above calculation.
To calculate the 90% confidence interval for the true mean, we can use the following formula:
Confidence Interval = Mean ± (Critical Value) * (Standard Deviation / √N)
Let's calculate the confidence interval step by step using the information provided:
1. Calculate the critical value:
Since we want a 90% confidence interval, we need to find the corresponding critical value. To do this, we can look up the z-value (standard normal distribution) for a 95% confidence level, as the remaining 5% is divided equally between the two tails. Using a standard normal distribution table or a statistical software, the critical z-value for a 90% confidence level is approximately 1.645.
2. Determine the standard error:
The standard error represents the standard deviation of the sampling distribution of the mean. It can be calculated using the formula:
Standard Error = Standard Deviation / √N
Since the standard deviation (StDev) is given as 1.467 and the sample size (N) is 23, we can plug these values into the formula to calculate the standard error:
Standard Error = 1.467 / √23 ≈ 0.307
3. Calculate the confidence interval:
Now that we have the critical value (1.645) and the standard error (0.307), we can calculate the confidence interval.
Confidence Interval = Mean ± (Critical Value) * (Standard Error)
Confidence Interval = 8.221 ± (1.645) * (0.307)
Confidence Interval ≈ 8.221 ± 0.505
The lower bound of the confidence interval is approximately 7.716 (8.221 - 0.505) and the upper bound is approximately 8.726 (8.221 + 0.505).
Now let's move on to the two assumptions necessary for the calculation:
1. Random Sampling: The sample of 23 customers should be selected randomly from the population of all customers at the shopping center. This assumption ensures that the sample is representative of the entire customer population.
2. Normal Distribution: The data collected on the distances travelled to reach the shopping center should follow a normal distribution. This assumption is crucial as we use the z-distribution to determine the critical value. If the data is highly skewed or does not follow a normal distribution, the confidence interval calculation may not be accurate.
Concluding statement:
Based on the given data, we are 90% confident that the true mean distance travelled to reach the shopping center is within the range of approximately 7.716 to 8.726 kilometers. This means that, on average, customers travel between 7.716 and 8.726 kilometers to reach the shopping center with 90% confidence.